Di Matteo was given the job for the remainder of the season following the sacking of Andre Villas-Boas in March and led Chelsea to success in both the FA Cup and Champions League. Despite the trophies it remains unclear if owner Roman Abramovich will give the former West Brom boss the job on a full-time basis.
"(A new manager could be appointed) any time over the next couple of weeks and certainly by the time we go on tour," he said.
"The players report back on July 4th or 5th and in an ideal world you want the coach in position by then."
Gourlay, who was appointed Chelsea's chief executive in 2009, believes Di Matteo's successful couple of months in charge mean he remains squarely in the frame, even though no discussions have taken place with Abramovich as to who should take the helm.
"It certainly would have made the discussions a bit different. The fact that he has the Champions League on his CV helps him enormously," he said.
"I know there has been speculation in the press and media about conversations with 'XYZ' - That has not been the case.
"We will sit down as a board of directors with our owner as quickly as we can and take our decision from there. We are going through that process now and then we'll sit down ... it's only been a couple of days (since the season ended)."
Meanwhile, Gourlay has backed Fernando Torres to replace Didier Drogba as the club's main goalscorer.
Drogba, who scored the winning penalty in Chelsea's Champions League final victory over Bayern Munich, is expected to move to China after revealing he will leave Stamford Bridge. The Ivory Coast forward has a prolific record at Chelsea, unlike £50million man Torres, but Gourlay has backed the 28-year-old Spaniard to get back among the goals.
"We've got Fernando Torres. And Fernando Torres is the man that we will go forward with and Fernando Torres will score us the goals," Gourlay said.
"Have no doubt that Fernando Torres is a world-class centre-forward and Fernando Torres will score goals for Chelsea. I'm not worried at all. We've just won the Champions League and Fernando Torres played his part in that. It's not all about goals. It's not just about the individual scoring goals, his teamwork - he is a very, very important player."