The 26-year-old has struggled to reach playtime at Anfield and could turn a finishing date to the Saints.
Liverpool are considering an offer from Southampton to loan Takumi Minamino for the rest of the season.
The Saints have contacted the Reds late on the final day of the transfer window as they look to strengthen their forward line.
No decision has been made with Liverpool at this stage, but sources have stated Amas that they would be open for a deal for the Japan international, who has struggled with first-team chances to come before this season.
Minamino has made 17 appearances in all competitions this season, but was included in the starting XI just seven times. He has started just twice in the Premier League.
The last of those starters, at Crystal Palace in December, saw his first league goal for the club, but he has only got six minutes of playing time since then – plus a 61-minute run. against the Aston Villa youth team in the FA Cup.
Southampton have moved on without getting a contract for Joshua King at Bournemouth, who are looking forward to joining Everton or Fulham. But with less than two hours left in the window, they will face a race against time to complete the turn, if Liverpool decide to allow it.
Liverpool have confirmed that the 25-year-old midfielder Ben Davies was signed from Preston earlier Monday completing a stunning £ 1.6 million ($ 2.1m) turnover.
They are hoping to secure another new defender before the deadline, with Ozan Kabak to sign on loan from Schalke, with the £ 18m ($ 24.5m) permanent transfer option in the summer.
Sepp van den Berg, the 19-year-old Dutch midfielder signed from PEC Zwolle in 2019, has joined Preston on loan for the rest of the season, and goalkeeper Liam Hughes, 19, has arrived from Celtic and will be part of U23 Founder Barry Lewtas.
A move is also being sought for Derby teenager Kaide Gordon, 16.
Liverpool, who are currently four points behind Premier League leaders Manchester City, will be next on Wednesday when they host Brighton. Four days later, they go face-to-face with Pep Guardiola’s side.