The 27-year-old joined the Hammers from Paris Saint-Germain in August 2022, penning a four-year contract, and he was a regular fixture in David Moyes's side last season.
Kehrer made a total of 38 appearances for West Ham in 2022-23, including 27 outings in the Premier League and a further eight in the club's triumphant Europa Conference League campaign.
However, the Germany international has since fallen down the pecking order at West Ham this term and has started only four of his 12 appearances in all competitions, with none of those starts coming in the Premier League.
While Vladimir Coufal and Ben Johnson have battled with each other for starts at right-back, Kurt Zouma, Nayef Aguerd, Angelo Ogbonna and summer signing Konstantinos Mavropanos have all featured regularly at centre-back over Kehrer.
A recent report claimed that Roma had been eyeing up a January move for Kehrer, but Monaco appear to have moved to the front of the queue for his signature.
According to journalist Fabrizio Romano, Kehrer will travel to Monaco on Thursday for a medical after the French club reached an agreement with West Ham over an initial loan deal worth €500,000 (£431,000).
Romano also states that an €11m (£9.5m) mandatory buy clause has been included in the deal, although reports from other media outlets, including Sky Sports News, claim that Monaco's agreement only contains an option-to-buy clause.
Monaco are said to be hopeful that Kehrer can complete his move to the club within the next 24 to 48 hours in time to make his debut against Lens in a Coupe de France fourth-round tie on Sunday.
Kehrer is closing in on a move to a Monaco side who currently sit third in the Ligue 1 table at the halfway stage, although they are not competing in European competition this term after finishing outside of the top five in the French top flight last season.
The defender may not be the only West Ham player on the move this month, as right-back Coufal has allegedly been left angered by the club's decision to not offer him improved terms on a new contract.
Coufal has played 24 times for the Hammers in all tournaments so far this season, but the Czech Republic international is due to see his contract expire in June, and it is thought that he could force a move in January if he does not pen fresh terms on a new deal.