The Roma and Netherlands midfielder is still recovering from a serious knee injury, but the Red Devils boss is prepared to bide his time and make a bid in January.
The 24-year-old has already developed quite a reputation on the continent thanks to his performances for PSV and then Roma, and his former Netherlands boss has rightly identified the midfielder as the priority target for his new United squad.
MORE: Biggest summer transfers | Soccer's beautiful fans
Van Gaal knows exactly what difference Strootman makes to any team, having changed his tried and trusted 4-3-3 formation to a 3-5-2 on the eve of the World Cup solely because of his star midfielder's absence. Given his intention to implement the same shape at Old Trafford, it's little wonder there is just one man for the job.
It is easy to see why Strootman is that man. From the early days in the Netherlands to his first season in Rome, he has stood out throughout his short career, imposing his game wherever he has been.
Having started out in the junior ranks at Sparta Rotterdam, relegated in 2010, it quickly became obvious that he was too good for the second tier. He did not hang around, either. Halfway through the following season he was snapped up by Utrecht, and, after making his Netherlands debut a month later, he was soon on the move again as PSV came knocking.
It was in Eindhoven that he really began to build his profile as a first-class midfielder, dominating the center of the park for the famous club. Even after the return of Mark van Bommel from AC Milan, a man who may have felt entitled to his old role following seven sparkling years away from the club, Strootman continued to boss matches for the Rood-witten, with the former Bayern Munich man forced to take up an unfamiliar role on the right of the midfield three.
Last summer his staggering early rise continued when he left PSV for Roma in a 17 million euro deal. So highly rated was he at Trigoria that the management handed him the No.6 shirt which had previously been retired for club legend Aldair. But rather than shrink under the expectation, Strootman quickly became a key feature of the new-look Giallorossi.
*2013-14 Serie A stats
Rudi Garcia's side hit the ground running with 10 successive Serie A wins as the club launched a rather unlikely Scudetto bid, and the Dutchman was at the heart of the positive approach which punctuated the team's season. In a side boasting the likes of Daniele De Rossi and Miralem Pjanic, quality ball-players in their own right, Strootman quickly became Roma's star midfielder.
With great passing ability and the vision to pick out a killer ball, Strootman was always likely to be a hit in the eternal city, but his eye for goal and imposing figure when not in possession helped to make him an all-round success in no time at all.
"For over €100 million we can sit around a table and think about it," said Garcia recently when asked about Strootman's availability, and while that was obviously in part to scare off suitors it also gave a reflection of the midfielder's impact. It may have also hinted at Ed Woodward's repeated insistence that Manchester United has money to burn.
The knee ligament damage which ended his season two months early and made him a spectator at this summer's World Cup looked likely to also keep him off the market this summer. But Van Gaal's increasing insistence that United make him the club's No.1 target and build its new midfield around him is no surprise.
|MAN UNITED LATEST
The Red Devils' difficulties in the middle last season were significant, and the Dutch coach has been quick to address the shortage in quality. But after being frustrated in their inquiries for some of Europe's most established names - there have been varying degrees of talks over Arturo Vidal, Angel Di Maria and Bastian Schweinsteiger - United could be ready to wait.
It could be a risky approach. Van Gaal has been filling last season's more maligned players - including Tom Cleverley - with renewed confidence during preseason, but with Michael Carrick injured and Marouane Fellaini on the way out, the Red Devils would certainly be light on numbers.
There is also concern over how Strootman will bounce back from his knee injury. Recovery from such serious ligament damage tends to go one of two ways, and as such it may pay for the 20-time English champion to wait and see how he fares. But while some have been ruined by knee complaints, United knows only too well that things can work out for the best - the huge number of goals scored by Ruud van Nistelrooy can attest to that.
But considering Strootman's tremendous talent even when not stretched to his physical limit, holding off now may be fatal in the longer term. This would be the equivalent of signing Vidal from Bayer Leverkusen before he was snapped up by Juventus.
Leaving United open to the possibility of missing out on a player of Strootman's quality and relative youth would be nigh-on foolish for a club which have so often been out-flanked when shopping for the world's top midfield talent.
Follow Kris Voakes on