The 23-year-old has struggled to live up to expectations since moving to Anfield from Newcastle for £35 million 18 months ago.
He is reportedly wanted on a season-long by AC Milan, and Rodgers said in The Daily Telegraph: "It's something I would have to look at, I have to be honest."
He added: "I'm not going to sit here and say I will never let anyone go on loan, then come in here in two weeks and a player's gone, and you're saying 'you said you wouldn't let them go'.
"There are many things to going on loan. Is it going to be beneficial for the club, that's the most important thing?
"Sometimes a player going out on loan - in general, not just Andy - can benefit the club in the long term.
"It gets them game experience, and they come back a better player, a more confident one. Certainly more so than if they've been sitting on the bench for the majority of the season.
"Andy's always going to be linked with clubs, whether he was here or not. I have spoken to him on his holidays, he knows exactly where he stands, but I have had no inquiries about him."