The 32-year-old midfielder is in the final year of the contract he signed back in 2020, while many have expected that he will not be offered new terms.
The former Bayern star is one of Liverpool's highest-paid players, reportedly on around £200k a week in basic pre-tax wages, but his game-time has been severely limited by injury issues in the last year.
Thiago did feature in 28 matches for the Reds in 2022/23. However, only half of those came as starts and he has not featured in a single minute for Liverpool this season as he continues his recover from a hip injury that has plagued him since the spring.
Despite his persistent fitness issues, Klopp is among those at Liverpool who want to see the number six handed a new contract, according to 90min. However, others at Anfield remain unconvinced.
The report claims that Liverpool are considering their options regarding the midfielder and that 'conversations have taken place about his future'.
It is added that some Reds officials are 'not wholly convinced' that re-signing Thiago would represent value for money, alluding to his his expensive wages and frequent injury issues.
90min claim that Liverpool will meet with Thiago and his team before the January window to 'assess the situation' but haven't ruled out a sale, amid interest from other clubs.
However, the former Barça star made it clear that he wanted to continue his recovery at Anfield and fight for his place in the team.
There was also a late flurry of interest in the player before the close of the transfer deadline in Turkey earlier this month, though that speculation ultimately came to nothing.
As well as Turkey, the Saudi Pro League and Spain's La Liga have regularly been touted as next possible destinations for the two-time Champions League winner.