The 30-year-old joined Liverpool for £70m in the summer of 2018 from Serie A outfit Roma, establishing himself as one of Jurgen Klopp's key men at Anfield.
Alisson has featured in 232 games for the Merseyside giants, helping them to Premier League and Champions League successes during his five-year period at the club.
The Brazil international only missed one league match for the Reds last season as they fought back from a poor start to finish fifth, securing Europa League football for the 2023-24 term.
According to The Mirror, Al-Nassr are looking to lure Alisson away from Anfield and European football this summer, as the Saudi Arabian side aim to win their division this season.
The Saudi Pro League club have already secured the services of notable players over the past 12 months, bringing in Cristiano Ronaldo following his departure from Manchester United and ex-Liverpool winger Sadio Mane.
It is understood that Al-Nassr are confident that they would be able to offer a financial package to Liverpool and Alisson that would be difficult to refuse.
Despite this, the report claims that Klopp's Reds have no desire to sell their first-choice goalkeeper in this window, with the Merseyside club looking for stability in defence following a host of midfield exits.
The departures of Jordan Henderson and Fabinho to Al Ettifaq and Al Ittihad respectively have reduced Liverpool's defensive-minded options in their engine room.
Former Reds club captain Henderson made his debut for Steven Gerrard's side on the opening weekend of the Pro League, with Ettifaq beating Al Nassr 2-1 courtesy of goals from Robin Quaison and Moussa Dembele.
The report adds that Alisson has not expressed any desire to leave Liverpool for the riches of the Middle East, although Luis Castro's men are optimistic that a deal can still be arranged before the end of the window.
Saudi Arabia international Nawaf Al-Aqidi is currently Al-Nassr's first-choice stopper, however the 23-year-old made an error in his side's defeat on Monday which may have thrown his starting spot into doubt.
Alisson was denied a first clean sheet of the campaign by Chelsea centre-back Axel Disasi on Sunday afternoon, with Klopp's Liverpool sharing the spoils at Stamford Bridge in their Premier League opener.
The 30-year-old is expected to start between the sticks once again for the Reds when they welcome Bournemouth to Anfield on Saturday afternoon - the hosts won the same fixture 8-0 during the 2022-23 term.