Black Cats winger Charis Marvrais spent the second half of last season in Germany.
Mavrias scores his only goal for Sunderland against Kidderminster
Mavrias, a £2.5 million Paolo Di Canio-era signing, has played just seven times for the Black Cats to date.
And the 22-year-old has since been loaned to former club Panathinaikos and Dusseldorf, where he spent the latter half of the 2015-16 campaign.
Mavrias while on loan to Panathinaikos
Despite scoring and assisting just once for the 2.Bundesliga side, Mavrias is wanted back at the Esprit Arena on a permanent basis, according to Funkel.
The Chronicle reports a £400,000 purchase option was inserted into his Dusseldorf loan agreement, but adds that the deadline for that to be exercised has passed.
And Funkel admits that the Germans' initial offer, which is said to be lower than the above figure, has been rejected.
“Sunderland have rejected our offer, which was probably too low,” said Funkel, as quoted by the Chronicle.
Then 1860 Munich boss Funkel (L) with Jurgen Klopp
“We have to think about it and I do not know how we will proceed. It is quite clear that there are economic limits and we are always looking for alternatives.
“Whether it works with Mavrias or not, I don’t know. Harry is very fast and can play attractive football. I think he is a really good player.”
Mavrias has made just one appearance for Sunderland under Sam Allardyce - a 14-minute cameo in January's FA Cup defeat to Arsenal at the Emirates.
The Mackems have allowed four players to leave this summer but are yet to make any new signings.
Mavrias during his most recent Sunderland appearance