We shall soon find out. On Wednesday, Rangers entertain Anorthosis Famagusta of Cyprus in the first leg of a qualifying tie for the European Champions' League. With Rangers' domination of the Scottish game complete, the pounds 4.5m outlay on Gascoigne was made with one aim only: to improve their disappointing record in Europe.
A game against the Cypriot champions might appear a formality, but the recent record of Scottish clubs in Europe is poor, and 12 months ago Rangers were knocked out at the same stage by AEK Athens. A similar result against the Cypriots would be a unthinkable disaster for the Glasgow club, and would immediately reduce Gascoigne's role to that of an expensive irrelevance (they always win the title anyway).
The signs are, however, that Gascoigne will make a difference. In a total of 141 minutes action against Steaua Bucharest and Sampdoria last weekend, he won over any Glaswegian doubters. A newly arrived player can rarely have had such a whole-hearted welcome from a set of supporters. Nor could he have wished for his new team-mates to try harder to help him settle in.
Gascoigne admits to having to mime when Rangers sing their victory songs, and he will surely be encouraged to leave any further displays of flute- playing to James Galway, but he has immediately looked comfortable in his new environment.
It helps, of course, that numerous English players have already blazed a trail to Ibrox. You could also argue that he is rather closer, culturally as well as geographically, to his Geordie roots in Glasgow than he was at Spurs. Most of all, however, he is a round peg in a round hole in a football sense.
Walter Smith, the Rangers manager, is too diplomatic to demean his other stars as bit-part players by admitting that Rangers' new playing system is purely geared to Gascoigne's needs, but in effect it is. With three men across central defence, he has players to cover for him when he gives the ball away, as his natural style of play makes it inevitable that he will on occasion.
The two wing-backs get into positions where he can pick them out with long passes from deep - a couple with the outside of the boot in the International Tournament were worth the price of admission on their own - and he has an abundance of midfield partners ready and willing to fetch and carry for him in midfield.
Most promising of all, though, is his burgeoning combination with Brian Laudrup. For two players previously unacquainted with each other, their understanding, particularly in the Steaua match, was uncanny.
The coach of Sampdoria, Sven Eriksson, is not given to fanciful verbiage or extravagant praise, but he described some of Gascoigne and Laudrup's combinations as "fantasy football" - and that from a man with Clarence Seedorf in a side that has only recently lost the services of Attilio Lombardo and Ruud Gullit.
Eriksson witnessed some of Gascoigne's struggles for fitness and form in Italy, but he expressed himself completely unsurprised by how good he looked in a Rangers shirt. "When he was fit in Italy, he played great games," he said. "He just needs to be in good shape and everything else is easy for him."
Compared with an injury-ravaged last three years, Gascoigne is in notably good shape now. Understandably, he tired towards the end of both matches, but Smith is confident enough about Gascoigne's fitness now that he would not have hesitated to play him for the full 90 minutes if there had been only one match that weekend.
Although it has always been fashionable to sneer at the infantile aspects of his personality, Gascoigne has been a hugely popular player with the fans of all of his clubs. Indeed, it is difficult to imagine him performing to anything like his maximum without that surge of public emotion behind him.
That will not be a problem at Rangers. Even a half-full Ibrox gave him the sort of response during the pre-match kickabout for which other players wait an entire career. When he actually did something eye-catching during a match - which was frequently - or when he left at half-time and at the end of his requisite 70 minutes, the adulation was deafening.
He already occupies a position in the affections of Rangers fans where he can do no wrong. When he said "they've taken to us", it was one of the bigger understatements of his career.
The true test of his value to Rangers, however, will not be in glorified friendlies or even in Scotish domestic football, which they can dominate comfortably enough without him. It will be in the European campaign which begins, albeit in a low-key manner, on Wednesday. With Gascoigne and Laudrup in tandem, Rangers' fantasies at that level can become realities.Reuse content