Brannagan has yet to feature in a first team matchday squad so far this season.
Brannagan was given some opportunities last season - making a total of eight first-team appearances in the process - but the 20-year-old's game time has been limited to action for the under-23s this time round.
The Echo reports that a move to Wigan Athletic fell through during the summer after the Championship side refused to accept a clause which would see the player feature in 75 per cent of their matches, as Jurgen Klopp wants any player that goes out on loan to get regular game time.
Given that Brannagan has not had a look-in with the first-team squad so far this season, some regular game time at a good level is needed to aid his development. Having got a taste for it last season, he would no doubt want more.
It will be interesting to see whether he does have a future with the Reds given that he has fallen down the pecking order - even behind younger players like Ovie Ejaria and Marko Grujic - but that could change with a promising spell playing away from the club.
Klopp seems keen to find solutions to roles in the side within the club before exploring the transfer market and this is a chance for Brannagan to impress should a deal be agreed.