I suggest you have a look at what (Stuart & Lister; Extinction chronology of the cave lion Panthera spelaea, Quaternary Science Reviews (2010), doi:10.1016/j.quascirev.2010.04.023) have to say on the topic, namely:

"It is remarkable that lion, in the form of modern Panthera leo, re-colonized a large area of south-eastern Europe in the Holocene, ca. 6 - 8 millennia after the extinction of the cave lion in Eurasia. The geographical range in the Holocene was limited to areas with open vegetation including the Ukrainian and Hungarian steppe, part of the former cave lion range, but they did not penetrate the forests of Central Europe (Sommer and Benecke, 2006)."

In view of the environmental characteristics of Italy at the time, as well as the lack of actual fossils, it is not likely that the "Lion reconquista" of SE Europe ca. 6.5-6.0 cal ka BP reached the Italian peninsula.

Otherwise, cave lions were gone from Europe around 12 to 13 cal ka BP, probably in response to the collapse of the Mammoth Steppe environment (op cit), as well as the increasing inavailability of Reindeer, which they favored as prey (see Bocherens, H., et al., Isotopic evidence for dietary ecology of cave lion (Panthera spelaea) in North-Western Europe: Prey choice, competition and implications for extinction, Quaternary International (2011), doi:10.1016/j.quaint.2011.02.023). They provide a pretty solid review of fossil sites and geochronology to go with that.

If you absolutely need Italian lions at that time as a plot device, perhaps you can cheat a bit and postulate a localized isolated population in the Alps, at high altitudes, where alpine meadows might perhaps be construed to be an environment amenable to those critters. But that would feel like grasping at straws.

We know that cave lions were around in Northern Europe around 35,000 ya, based on artwork dated to the Aurignacian period. A famous example is the Lion Panel at Chauvet, link here:

http://www.bradshawfoundation.com/chauvet/panel_of_the_lions.php

As to when they died out, historians aren't sure. There are two primary issues- one, are the lions mentioned in ancient sources actually cave lions (Panthera leo spelea vs. P. leo europea)? Descendants of cave lions? Maned lions migrating from Africa (P. leo)? This blog post has some of the relevant quotations:

http://www.strangehistory.net/2013/06/29/the-last-european-lion/

If the ones in literature are descendants or cousins to cave lions, a case could be made for cave lions existing in Europe until at least the Roman period. This is unlikely because the lions in recorded sources appear to be much smaller than cave lions in the fossil record, and genetic studies have indicated a species split, but it is (remotely) possible. See here for genetic study: http://www.sciencedirect.com/science/article/pii/S1055790303003300

Two, we don't know exactly why cave lions died out. These cuties date to approximately 10 kya and were preserved in permafrost so we know what cave lions ate, what they looked like, and where they lived, but not what made them extinct: http://www.dailymail.co.uk/sciencetech/article-3291448/The-face-10-000-year-old-cave-lion-cub-Two-incredibly-preserved-prehistoric-babies-discovered-Siberian-permafrost.html

This study indicates that cave lions disappeared around the same time that reindeer populations disappeared, and current reindeer habitat is inconsistent with supporting cave lion populations. http://www.sciencedirect.com/science/article/pii/S1040618211001182

Using that information, you could reasonably say that a cave lion would be present as long as reindeer are present. Reindeer probably left northern Italy when that area ceased to be tundra, no less than 15-20 kya.