Vieri trained with Rovers in Austria last week and is also due to join up with Allardyce's squad in Italy this week.
He is then expected to return with the club to England as the former Italy international looks to secure a move to the Premier League.
Allardyce accepts the 36-year-old's current level of fitness is below standard, but he wants to give the former Inter Milan and Juventus man a chance to clinch a deal.
"Vieri will follow us on to the next session away, which starts on Wednesday, then join up with us back here," said Allardyce in The Sun.
"Obviously, he is way behind the rest of the players and we will be very patient and supportive with him.
"We'll see how long he wants to give it to see if he can get to the level he needs to play in the Premier League.
"If he manages that, what a player. We will decide together, me and him. We will take as long as it needs, hopefully to get there in the end."