The Egypt international moved to Liverpool FC in a £39m deal from Serie A giants AS Roma in the 2017 summer transfer window.
Salah has scored 28 times in 30 games in the Premier League so far this season after his four-goal haul in a 5-0 win over Watford.
The 25-year-old has netted a further eight times in cup competitions to help Liverpool FC reach the Champions League quarter-finals.
The African forward has been linked with a potential move to La Liga giants Real Madrid in recent weeks.
Liverpool FC have a record of selling their best players to FC Barcelona and Real Madrid over the past nine seasons.
Just two months ago, Philippe Coutinho completed a £142m move to FC Barcelona in the 2017 January transfer window.
Former Liverpool FC striker Bellamy believes Salah is suited to the Reds’ style of player under Jurgen Klopp.
“The way Liverpool play suits him,” Bellamy told Sky Sports. “He has a manager who is perfect for him.
“I’d offer him a new contract straight away. Make sure he is rewarded. Bump it up even more.”
Liverpool FC finished in fourth place in the Premier League table last term to secure a return to the Champions League.
The Reds signed Netherlands international Virgil van Dijk in the January transfer window.
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