Schalke and Werder Bremen this summer.
The Everton boss’ future on Merseyside is in doubt, with the Scot yet to agree to a new contract. His current deal at Goodison Park ends in June.
And Moyes’ recent revelation that he has ambitions to manage abroad – and, specifically, in Germany – is triggering serious interest from the Bundeslega duo.
Schalke are looking for a long-term replacement to former manager Huub Stevens, who was sacked last month after a run of poor results left the club outside the top six positions.
Youth coach Jens Keller has taken charge on a short-term basis and Moyes, 49, now appears to be high up on Shalke’s list of potential candidates.
A source close to the club told the Daily Star Sunday: “Moyes is highly regarded in Germany and Schalke are definitely monitoring his situation at Everton.
“If he ends up leaving there in the summer, they will be at the front of the queue.”
But the seven-time German champions will face competition from rivals Werder Bremen for Moyes’ signature.
The North-German side are struggling in the Bundeslega this season. They currently occupy 12th place, 20 points behind league-leaders Bayern Munich.
It is thought that their long-serving coach Thomas Schaaf will leave the Weserstadion after 14-years at the club, paving the way for Moyes, whose managerial style and high-tempo passing game have already seen him favoured by the Werder board.