The Phobians, who have failed to win any major trophy since 2009, reassigned Wellington as head coach in January following the departure of Frank Elliott Nuttall.
Hearts currently sit 13th on the top-flight table with 17 points, having won just one league match in their last seven fixtures.
Youth team coach Seth Hoffman has been appointed as caretaker and will be in charge of Hearts' preparations ahead of the second round.
"The Board of Directors of Accra Hearts of Oak Sporting Club Limited has, at its meeting held on Thursday, 7 June, 2018, terminated the contracts of coaches Henry Wellington and Jerry Adjei with immediate effect," a club's statement read.
"Coach Seth Hoffman, currently the head coach of Hearts of Oaks Under-20, Auroras, has agreed to take temporal charge of the first team duties and will serve as interim head coach until a substantive head coach is appointed.
Also, Coach Nii Doodo Odoom has been re-assigned to the Auroras.
"Hearts of Oak wishes to thank Henry Wellington and Jerry Adjei for their services and wish them well in their future endeavours.
"Meanwhile, the Board will continue with the active search for a Director of Football and expect to have an announcement on that appointment in the coming weeks. At such time, the club will also name a substantive head coach.
"The Board would like to ask for continuous support and collaboration of all stakeholders for Coach Hoffman, his backroom staff and the entire playing body," the statement added.
According to Goal's sources, former Medeama coach Tom Strand's name has popped up again despite the club denying interest in his services.