Ally McCoist's side have seen off Hibernian and Clyde so far in the competition and will face East Fife in the last eight.
The Gers have missed out on the silverware in the last two seasons, but former Ibrox boss McLeish said: "It's a great occasion for East Fife but also a pretty tricky game for Rangers.
"It's a cup tie and I know that Ally McCoist wants to win every competition that he's in - he's a real competitor.
"Rangers lost out to Raith Rovers in the final last year but I'm sure Ally McCoist will want to win the Petrofac Training Cup this time around.
"I think Rangers will have learned some lessons from last year and there will be pressure on them to win - it's there for the smaller teams to do a giant killing but I do fancy Rangers this year.
"I think the game against Clyde will have given them a lot of confidence."