"It's not easy to reinforce Juve, but we have some very clear ideas and it's no secret that we are looking for an attacking midfielder and a defender," Allegri told reporters on Monday.
"Sneijder has tremendous technical quality and would complete an already very strong part of the team, creating competition for places.
"He wouldn't be able to play in the Champions League but we have to be realistic and see who is out there."
Sneijder, 30, spent three-and-a-half seasons in Serie A with Inter Milan before joining Galatasaray two years ago.
Juventus, who host struggling Inter Milan on Tuesday (2000 GMT), start the year three points clear of AS Roma at the top of Serie A, although they were beaten on penalties by Napoli in the Italian Supercup match in Doha on Dec. 22.
“The first game of the year is always a strange one. The team that stays the most focussed and alert will emerge victorious," Allegri said.
"It won’t come down to physical condition or technique, it will be mindset and mindset alone."
“The lads have returned from their holidays fired up and raring to go.
"We won’t be suffering a hangover after Doha. I always like to win and unfortunately it didn’t happen in Qatar, but we played well and the game was only settled after 15 penalties were converted during the shootout."
(Writing by Brian Homewood in Berne, editing by Ed Osmond)
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