The Argentine arrived at the DW Stadium just last summer, but has found it hard to cement his place in the team and has failed to find the net in the Premier League, scoring just once in 12 appearances this season.
After speculation of a loan move to Bari, Genoa have now entered the frame for the former Boca Juniors striker.
"I like Boselli, he is our aim," Genoa owner Enrico Preziosi told Primocanale. "He's an interesting player in perspective.
"He played with Rodrigo Palacio at Boca Juniors, they know each other.
"We will try to seize the best opportunity."
Genoa are currently sitting 13th in the Serie A table following a mixed start to the campaign, and Preziosi is hoping to revamp the side this month.