The Reds have been searching for a wide attacking option this summer following Mohamed Salah's departure, and identified the 22-year-old Gambia international as a priority target.
The Athletic reported that Liverpool's opening £50 million offer was turned down last week, before Sky Sports reported that an improved £60 million bid met the same fate.
According to journalist Sacha Tavolieri, Minteh and his representatives have since made clear to Brighton that the player wants the move to go ahead, adding pressure on the Seagulls to reconsider their stance. Liverpool are said to be preparing a third bid in response.
Minteh scored three goals and provided four assists in 34 league appearances last season, primarily on the right flank but also covering on the left during Kaoru Mitoma's absences, and is under contract at Brighton until the summer of 2029.
The winger joined the Seagulls from Newcastle United in the summer of 2024, and has established himself as one of the Premier League's most sought-after attacking talents outside the traditional elite clubs, making him a difficult player for Brighton to release without a fee that reflects his rising value.