Although he spent £150million on six new players last summer, Van Gaal has been assured that he will be given sizeable funds to improve the squad in the close season if he deems it necessary.
With Michael Carrick turning 34 this summer, and the jury still out on Daley Blind, the United boss is concerned he does not have enough cover in front of the back four so he has made a dynamic midfielder his priority and Schweinsteiger certainly fits the bill.
The 30-year-old has won the Bundesliga eight times, lifted the Champions League with Bayern in 2013 and was part of Germany's World Cup winning squad last summer.
The midfielder has made over 300 appearances for the Bavarian club since he joined as a 14-year-old, but his contract expires next summer and talks have reached an impasse.
Press Association Sport understands the midfielder is on Van Gaal's transfer shortlist and the United boss is hoping to sign the player this summer.
Van Gaal used his personal relationship with Memphis Depay to persuade him to sign for United and the 63-year-old hopes that will be the case with Schweinsteiger.
Van Gaal coached the German for two years at Bayern
He converted the 30-year-old from a winger into the defensive midfield position where he plays now, and together the pair won the Bundesliga and the DFB Pokal.
Senior sources at United admit Schweinsteiger is more likely to remain at Bayern, but that has not stopped them privately expressing their interest in the Germany skipper.
United, meanwhile, confirmed on Thursday that the second match of their pre-season tour of the United States against the San Jose Earthquakes will take place at 62,467-capacity California Memorial Stadium.
United will take on the Major League Soccer side on July 21.
Van Gaal's team will kick off the tour against Mexican side Club America in Seattle
After playing the Earthquakes, they face Barcelona in Santa Clara and Paris St Germain in Chicago as part of the International Champions Cup, which they won last summer after beating Liverpool in the final in Miami.
Source : PA
Source: PA