Courts has been recently linked with a move to Croatian side Rijeka, while reports from The Courier suggest that Hungarian outfit Honved Budapest and some English clubs are also keen on the 41-year-old.
Former Hibs and Sunderland manager Ross is expected to be appointed as Dundee United’s new manager, with the Terrors having already identified “a preferred candidate”.
Courts led United to their highest-placed finish in nine years and secured European football for the first time in a decade.
He also gave 16 academy graduates first-team experience, with Archie Meekison, Kieran Freeman and Ross Graham earning Scotland Under-21 call-ups.
He told the club's official website: "I'm immensely grateful to everyone at Dundee United for the opportunity they gave me to manage such a prestigious club.
"Tony Asghar (sporting director) took a chance on me and I had terrific backing from the owner Mark Ogren, the Exec Team and from my backroom staff such as Liam Fox and Adam Asghar.
"A special thank you must also go to the players who have been a joy to work with and a pleasure to lead.
"They give maximum effort to everything they do for the club and I hope they have continued success for next season and beyond."
Courts, who took over United last season after a spell in the club's academy, added, "Without their support and the backing of the fans, we wouldn't have achieved our fourth-placed finish and qualification for the Europa Conference League.
"It's not an easy decision to move on from Dundee United but I believe my time with the club has helped me grow and develop immensely, both personally and professionally and I feel ready for my next challenge."
The statement on United's website confirmed that after Courts had expressed "a desire to explore other options away from Tannadice, we have now reached an agreement with Courts that will see him leave the club with immediate effect".
Asghar anticipates making an announcement on a new boss in the coming days.
He said: "I'd like to place on record our gratitude towards Tam Courts for the service he has given Dundee United both as head coach and in his previous role as head of tactical performance.
"We made a brave appointment last year by promoting from within but it was a calculated risk that clearly paid off.
"We enjoyed a lot of success on the field last year and qualified for Europe. This is testament to the culture we have created throughout the club."
Ross’ name was linked with the Dundee United position last week when Rijeka’s interest in Courts was first reported and the 46-year-old is the red-hot favourite to be the next manager.
A former Dundee trainee, Ross has been out of football management since being sacked by Hibs in December last year. He has been in the frame for many jobs since, including Aberdeen, Kilmarnock, St Mirren and most recently the Dees, but has been waiting patiently until the right opportunity arises.
Ross guided Hibs to a third-placed finish in the cinch Premiership and the Scottish Cup final during the 2020/21 campaign. He also has experience of managing in Europe, with Hibs’ run in the Europa Conference League being ended, ironically, by Rijeka last season.