Were there efforts to somehow resurrect the ancient Roman Republic into the modern world stage? No. Were there efforts to shape the unification of Italy towards the creation of a modern republic with ideas and influences taken from the Roman Republic? absolutely. Two of the "fathers" of Italian unification, Giuseppe Garibaldi and Giuseppe Mazzini, were both advocates of unifying Italy as a republic, in line with the ideals of the French revolution blended with Italian republican traditions that existed in early-modern Florence, Pisa, and Genoa, which in turn borrowed influence from the ancient Roman Republic.

Republicanism has strong roots in the Italian peninsula all through its history since the Roman era. Niccolò Machiavelli, an early-modern Florentine statesman, was an avid advocate for Florentine republicanism that was influenced in no small part by his love of the institutions of the ancient Roman Republic, which he wrote extensively about in his Discourses on Livy. Dante Aligheri, widely considered to be the greatest author in Italian literature, was also very much engaged in the politics of the Republic of Florence as a White Guelph, a political faction in Florence which opposed Papal influence and supported increased independence from the Holy See. Italian Republicanism was very much rooted in the culture throughout the Tuscany and Romagna regions in particular, but this influence was in conflict with the tradition of Monarchism which had a strong base in Piedmont-Sardinia and Naples, two much more powerful political forces on the Italian peninsula by the time of the Risorgimento.

The Napoleonic Wars and Napoleon's attempts to politically unite Italy under a French sphere of influence did much to break the divisions between the Italian states of the early-modern era and create a sense of national identity among Italians on the peninsula, but this sense of nationalism was mainly limited to the educated classes/Intelligentsia of Italy, who formed social circles to discuss the politics of Italian unification as well as resistance movements to unite the country by force. The main resistance movement of the early Risorgimento was the Carbonari, which sought to unify Italy through revolution. Among its members were Garibaldi and Mazzini, both advocates for popular democracy in Italy who fought in a failed revolution against the Two Sicilies in 1820, and another failed campaign against Austria in 1830 in Modena. The Carbonari were eventually defeated, but their ideals of establishing a constitutional government in Italy lived on, mainly through a successor movement, Young Italy. Young Italy, with Mazzini as its founder and Garibaldi joining shortly afterward, continued to push for a united republican Italy with increasing support among the educated classes. With growing support for constitutional governments reaching a boiling point in 1848 as part of a European wave of revolution against the political status quo across the Continent. King Charles Albert of Sardinia ended up giving concessions to these revolutionaries by adopting a Constitutional Monarchy in Sardinia-Piedmont.

Mazzini and Garibaldi also never lost their interest for forcing an Italian republic into existence through revolution. In 1849, Mazzini, Garibaldi, and another Italian Republican, Aurelia Saffi, succeeded in overthrowing the Papal States and establishing a republican government in Rome, forcing Pope Pius IX into exile. A republic was declared and a constitution was written, but the French Empire under Napoleon III came to the support of the exiled Pope. French troops recaptured Rome, placed the Pope back in power, and negotiated for Garibaldi, Mazzini, Saffi, and 4000 other Republican supporters to leave Rome with their lives.

Although there was a strong sense of Republicanism in Italy, failed rebellion after failed rebellion established that the Italian republicans simply didn't have the political support or powerful connections necessary to unify Italy, while monarchist governments, namely the Kingdom of Sardinia-Piedmont, under Prime Minister Camillo Benso, Count of Cavour, did have the political connections necessary to obtain international support for Italian unification, namely from French Emperor Napoleon III, which I go over in this answer over here if you're interested. The Count of Cavour politically allied with Mazzini and Garibaldi in unifying Italy (though Mazzini was eventually muscled out of the picture by 1856), but under a banner of nationalism to establish a common identity among Italians rather than republicanism. Italy would eventually become united, but under a Monarchy ruled by the House of Savoy, albeit with a constitution influenced by Italian republicans with influences from Ancient Rome, such as the Italian Senate.

As for the construction of a Roman Empire, one can easily look to Italy's fascist dictator, Benito Mussolini, who did very much liken himself as "one of the Caesars" with an expansionist foreign policy centered on dominance of the Mediterranean Sea, as well as the revival of Roman iconography, notably the Roman Eagle and the Fasces, as well as reviving Roman terminology such as calling the Mediterranean "Mare Nostrum" (Our Sea). Mussolini's expansionist ambitions of Italy ultimately led to his defeat in WWII by the Allies, along with the downfall of the Kingdom entirely.

Italy wouldn't become a full republic until the aftermath of WWII, with a referendum held in 1946 between keeping the monarchy or establishing a republic ending in a victory for the republicans. While Italy didn't resurrect roman offices verbatim like consuls, praetors, or quaestors, the Italian constitution is very much influenced by the ancient Roman republic in its adoption of civil law that the Roman Republic pioneered (and more accurately codified by Eastern Roman Emperor Justinian), and the establishment of a Senate.

Sources:

The Pursuit of Italy: A History of a Land, Its Regions, and Their Peoples by David Gilmour

Risorgimento: The History of Italy from Napoleon to Nation State by Lucy Riall