The duo have a lengthy history of working together having first joined forces at Ajax, Koeman then taking him to Southampton while at St Mary's in the summer of 2015.
Koeman raised a few eyebrows when he made fellow Dutchman Stekelenburg his first signing as Everton boss and caused even more of a stir when naming the 34-year-old as his number one goalkeeper for this season.
Koeman's faith in Stekelenburg was firmly rewarded on Saturday though when he saved two penalties at Manchester City and the Netherlands international hopes he can produce more man of the match performances while at Everton.
"I'm thankful for that [Koeman bringing him to Everton]," Stekelenburg told the ECHO.
"He asked me last year to help him at Southampton because of an injury to the first goalkeeper. I had a good relationship with Koeman in my Ajax period so, of course, I am thankful he has brought me here but in the end you have to show it because if you don't show it....
"Ronald gave me a chance to play at Southampton because I want to get my games coming and I played 25 games so it was not a bad season playing-wise.
"And so as soon as he came here and asked if I wanted to come, there was no question about it."