Dorrans moved to Scottish Premiership giants Rangers from Norwich City in the Championship in England in the summer transfer window.
Wallace, who is also 30 years of age and operates as a left-back, has revealed that he tried to convince the former Livingston and West Bromwich Albion midfielder to make the switch to Ibrox last summer.
“After the disappointment of losing the Scottish Cup final a few of the families were over there on holiday — and so was Graham,” Wallace told The Scottish Sun. “Even at that point, he would have walked it to sign for us.
“It couldn’t quite happen then but thankfully it’s happened this summer. That sums the boy up — he was just desperate to come back and play for his boyhood club.”
“It was great that we just happened to be on the same holiday that year because Kenny Miller and I could get the conversation going with him,” added Wallace.
“Believe me, we did all we could to get him here even sooner! He stayed at Norwich for another year but it’s brilliant to have a player who really wanted to be at Rangers.”
Rangers kicked off their 2017-18 Scottish Premiership campaign with a 2-1 victory over Motherwell away from home at the weekend.
Dorrans scored both the goals for the Gers at the Fir Park Stadium on Sunday afternoon, first in the fourth minute and then from the penalty spot on 58 minutes.
The Scotland international midfielder has revealed that his teammate Josh Windass wanted to take the spot kick, and believes that he will be on penalty duty this season.
“There weren’t really any nerves when I went to take the penalty, Josh came in and wanted to take it but Helder Baptista spoke to me before the game and said I was to take it,” Dorrans told The Scottish Sun.
“I hope I’m on penalties from now on! For as long as I can keep putting the ball in the back of the net than I am sure I will be.”