No. Fascism is a XX Century thing (there have been, of course, legitimate attempts to look back for certain traits of "fascism" in previous political formations - the usual candidate being Napoleon III in France). For Italian Fascism specifically - the one which introduced the usage of the fasci littori and turned them into an official symbol of the Regime - we are sescribing a political movement which appeared after the Great War, in 1919, and gained power in 1922.

At the time, the "fasces" - as defined by Merriam-Webster online dictionary:

a bundle of rods and among them an ax with projecting blade borne before ancient Roman magistrates as a badge of authority

had already become one of their distinctive symbols. The fascists had run, rather unimpressively, for the 1919 national elections and adopted the symbol of the fascio for two of their three semi-independent lists. Thise lists had chosen the “Roman Eagle" (Rome) and the “Fascio dei Littori" (in Milan and Turin) respectively.

Initially this was more of an attempt to characterize their lists with “more precise patriotic content” and with “a memory of glorious events of our past, recent or remote” - according to the contemporary report compiled by the bureau of statistics. And only in subsequent times, during its transformation in Regime, Fascism begun to "institutionalize" its relation with ancient Rome, giving large impulse to scholarly research and archelogical investigation for the purpose to establish the "accuracy" of the Fascist restoration of romanity.

I believe there are a few pictures of Mussolini actually going around with the twelve lictores carrying the fasci.

The history of this process is a rather curious one, since the choice of the name Fasci di combattimento, the identification with the symbol and the relation with ancient Rome appear to be more or less incidental. In fact, if the English word "fasces" has a rather precise connotation, the Italian one was used mostly to refer to something else.

Nowadays, within Italian language, the use of the word fascio has often fallen in line with XX Century events (exception made for a few technical terms, in anatomy and mathematics for example, and a few frozen idiomatic expressions, the most notable being fare di tutta l'erba un fascio, more or less analogous to “tar the whole bunch with the same brush”) so that the relation with the fascist movement has become an objective one – while the various related terms, fascia, fascina, fasciare, fascicolo, etc survive free from the association with fascism and its symbology. With that in mind (and considering the vast amount of related terms, which have no equivalent relations in English for the voice “fasces”), it's easy to see how, as of early XX Century, fascio wouldn't really work (except for the most literary oriented) as an obvious reference to the symbol of authority of the Roman magistrates.

The vastly attested main use of the word was just what the first part of the definition gives us, a more or less exact analogue of "bundle": "a group of things fastened together for convenient handling ".

In fact, of the many possible meanings given to the word, the "first" Italian dictionary – curated by N. Tommaseo and which begun publishing in 1861, in Turin by G. Pomba – the Roman "bundle of rods with ax" is listed as the eighteenth, being noted as “plural only” (that would be fasci) and marked as "ancient historical". The first one is just:

Qualunque cosa accolta insieme o legata, di peso tale che uomo possa portarla.

Whatever thing, collected together or bound, of such a weight that a man can carry it.

This was also the same first acception included in the 1806 edition of the Dictionary of the Accademia della Crusca - dedicated to the viceroy of Italy, Eugenio Beauharnais (and despite the frequent occurrance of the "republican fascio" in the coinage of the French republics certainly making it more popular). That of things fastened together (in this fashion it was used for instance by V. Alfieri, presi a fascio, “taken collectively” in 1803), but also, in an extension of meaning, to signify a collection, a group, a sizable amount.

A second – and no longer extant, but well attested in literature, as many other examples of uses are given – was that of fascio for “burden”, as in “his joke is easy, his burden is light”. Additionally, the negative acception is listed – which survives in the composite form sfasciare - in the form mandare in fascio, to “cast into ruins”. And further away from the original meaning, an older version of the common saying fare di tutta l'erba un fascio - which, rather than a tendency to apply collective and broad strokes judgment onto others, stressed the negative moral connotation inherent within the inability to discern good from evil, so that the expression was tantamount to “living a dissolute life”.

Still above the Roman history acception, an idiomatic equivalent of “excuses” (as in “you always find an excuse”) is listed in Tommaseo, as well as in previous dictionaries.

It is noteworthy that the dictionary notes the existence of the invariant plural for those fasci of rods borne before the roman magistrates (with the possible “addition of laurel wreaths”) but makes no mention of their recent usage during the republican years, no reference to the use of the word for republican symbology (which may be an intentional omission, even if the “Republic” voice does include a reference to a contemporary, absolute, use of “republic” to signify the French one of late XVIII Century – and the symbol was used, as we'll discuss later).

Antonio Lissoni's Dictionary of Italian Phraseology – in its second edition of 1836 – lists the meanings mentioned above, but gives I fasci de'littori, rather than the simple fasci, as a fixed expression (and notes again on the possibility of adding laurel wreaths), which is the closest to a mention of fascio littorio before the 1920s I was able to find. Similarly a 1828 dictionary printed in Padua gives fasci as a “historical term”, analogue to the latinfasces, and both as plural invariant.

All this to say that fascio was a fairly common, and ordinary, word as of early XX Century – and that a reference to Roman antiquity in the form of fascio littorio would have been recognized in its use only within a very specific and narrow context, such as Ettore de Ruggiero's Dizionario epigrafico di antichità romane ("Epigraphical dictionary of Roman antiquities" [1886] and later editions). At the same time, the word did in fact lend itself to a rather obvious “political” interpretation, and with such a connotation it is fair to say that it had entered the public mind somewhere around the beginning of the process of Italian unification.

In 1815, it was used to symbolize the act of collecting and bringing together the different peoples of Italy in the unpolished last verses of A. Manzoni's unfinished (and unpublished until 1848) celebration of G. Murat's Proclama di Rimini, appealing to the Italians uniting for the “Independence of Italy”.

With Him, o lord of the Italian fates / those scattered rods you'll collect from the ground / and make of them a fascio in your hand

And in such obvious acception, it survived – and to some extent thrived as a political handle

Meanwhile the actual fasci littori - or something which looked like them – had begun reappearing after an undetermined period of obscurity as a symbol connected to the French republican ideas during the last years of the XVIII Century. The presence of those same symbols in Italy finds certain (and more “popular”) evidence in the coinage produced by the short lived Italian republics of 1797-99 – for instance, the Una Lira, Due Lire and Quattro Lire of the Ligurian Republic, with shield and Phrygian cap, surrounded by laurel and palm branches, of 1798; and the subsequent higher denominations, in gold, up to Novantasei Lire, with the fascio surrounded by laurel branches, and the words “in unity strength”. Analogous symbolism was chosen for the Baiocchi of the Roman Republic, with cap and oak (or laurel) branches, and the plain Baiocchi, with cap, of the Ancona Republic, also of 1798. Notable – but without ever circulating – the designs for the new emissions of 1803 made in Rome, with the introduction of the eagle holding the fascio on the obverse, and geometric compositions of two or three fasci on the inverse. Similar symbols were chosen for the Piedmontese Republic in 1799, this time with examples in the Mezzo Scudo and Quarto di Scudo, where the fascio, with cap (and oak wreath on reverse), is held straight by a figure of Armed Liberty. The Neapolitan Republic went all in with Liberty holding both the lance with cap and the fascio, while pressing her foot over the Bourbon Crown, in their Sei Carlini and Dodici Carlini of 1799.

Notably, the symbol seems to vanish during the following Napoleonic domination, when the design was either entirely changed to reintroduce heads on the obverse, or at least replacing the fascio with a single rod or lance (still with cap, for instance in the Five Francs of the Subalpine Gaul Republic of 1802).

You're thinking about things in reverse here. The Italian Fascists were consciously apping the symbolism of Ancient Rome. Although focused more on the Roman Salute than anything else (which is of particular interest since it was a false genealogy) this earlier answer of mine might be of interest for you.