The 20-year-old looks to be staying at Anfield following an unsuccessful approach from Wiagn Athletic. Middlesbrough are also said to be keen.
Gary Caldwell's side were reportedly willing to pay in excess of £1 million for the 20-year-old, but were told he is not available either permanently or on loan.
Wigan Athletic manager Gary Caldwell
The report claims Brannagan has also attracted interest from the Reds' Premier League rivals Middlesbrough, who like Wigan are newly promoted this season.
The Salford-born starlet made nine first-team appearances last term, the last of which, in the final day draw at West Bromwich Albion, triggered a re-negotiation clause in his contract as it was his fifth start.
And according to the Echo, talks are expected to begin over the coming weeks which could lead to Brannagan signing an improved long-term deal.
The Reds' academy graduate is set to feature heavily in Jurgen Klopp' side's pre-season campaign, with Emre Can and Joe Allen still at the Euros and Jordan Henderson and James Milner having not yet returned.
Wales' Euro 2016 explouts continue to push back Joe Allen's return to club football
Meanwhile, another youngster, Sergi Canos, is heading for talks with Championship side Norwich City after a £2.5 million offer was accepted by Liverpool, according to the Echo.
Former Brentford loanee Sergi Canos could be heading back to the Championship