Both still employ full-time managers in Tim Sherwood and David Moyes, respectively. Nonetheless, the continued scrutiny of their work, combined with the current Netherlands boss's availability post-World Cup, will ensure speculation continues until the involved parties confirm related negotiations.
In the event of either club dismissing their current manager—which, according to a Sky Sports report earlier this month, is just about a certainty at Tottenham—the experienced and successful Van Gaal is as close to an ideal replacement as is likely to be available.
The Mirror's Simon Mullock speculated over the weekend—via quotes from ex-Holland and Nottingham Forest striker Pierre van Hooijdonk—that Man United forward Robin van Persie might only stay at Old Trafford if his national team boss is appointed.
Intriguingly, Mullock wrote, "Dutch sources claim the 62-year-old is reluctant to give Tottenham an answer on an offer to take over at White Hart Lane only because he expects a vacancy to arise at Old Trafford."
Moyes' first year in Manchester has not gone as well as hoped. Last season's Premier League champions currently sit outside of the top six, 23 points behind first-place Liverpool.
Despite underwhelming results and performances, Moyes appears to retain the backing of his superiors.
Upon the Scotsman's hiring last May, United board member Sir Bobby Charlton spoke to the club's official website of "the long-term" and building "teams for the future as well as now" under Moyes. In an interview with BBC Sport last month, he admitted the Red Devils had played "really, really badly" this season, but he remained "absolutely certain that we picked the right man."
Besides the likelihood that Manchester United will not be looking for a new manager, Van Gaal would be wise to opt for North London regardless.
EDIT: As covered here by Bleacher Report's Timothy Rapp and announced at 8:30am, Tuesday morning, David Moyes has indeed been sacked by Manchester United. A further reminder that the word of people in football is about as valuable as counterfeit money.
The criticism of Moyes' troubled season has underlined the expectations at Man United (though it should be noted that many supporters have been more accepting of the difficulties following in Sir Alex Ferguson's legendary footsteps).
Having enjoyed success with three of Europe's biggest clubs in Ajax, Barcelona and Bayern Munich, Van Gaal would not shrink from such expectations at Man United. Still, the unhappy end to his time in Munich should make him think twice about taking on a club job of such scale again.
A stuttering Bundesliga campaign in 2010-11 for the reigning champions saw him sacked earlier than his previously agreed summer leaving date. As Raphael Honigstein of Sports Illustrated described it at the time, Van Gaal was a casualty of a "culture of enormous pressure."
The Dutchman was not entirely blameless, at least not according to then-Bayern president Uli Hoeness.
"Problems were created which were totally unnecessary and which have ripped the club to pieces," Hoeness said, via BBC Sport. "And to say that he had the players behind him was a myth."
United's hierarchy would be far less hands-on than Bayern's, but the two clubs' similar DNA—formed by numerous successes both recently and in the past—could mean Van Gaal would have to battle to implement ideas on a team whose many egos might be similarly resistant to his abrasive manner.
Tottenham are not a club without their own expectations. The demands for top-four football from the club's hierarchy played a large part in the departure of Andre Villas-Boas in December and are likely to do the same for Sherwood.
Spurs are in relatively good shape talent-wise, so the proven Van Gaal would have as good a chance as anyone to produce results tolerable (or better) to chairman Daniel Levy and Co.
The likes of Emmanuel Adebayor, Christian Eriksen, Hugo Lloris and Jan Vertonghen cannot point to medals and trophies won with Spurs in the way players like Michael Carrick, Wayne Rooney or Van Persie could with Manchester United. That leaves them less likely to think or tell Van Gaal that they might know better.
In Spurs, the 62-year-old would find a challenge similar to that which he took on at AZ Alkmaar—a club with considerable tradition but struggling to push on sufficiently in their attempts to match past glories. He would almost certainly be afforded sufficient control to implement his ideas and make changes where he sees fit.
Despite Spurs' issues this season, particularly coming up against the Premier League's current top four, Van Gaal would have plenty to work with already. In addition to the aforementioned players, Spurs have their best group of young talent (Nabil Bentaleb, Harry Kane and Andros Townsend among others) in several generations. Summer 2013 signings who have struggled this summer will also have benefited from a year's football in England under their belts.
Manchester United have a good core of talent too. Other notable players include top-class goalkeeper David de Gea and one of Europe's best creative midfielders in Juan Mata. Should he see fit, Van Gaal would have scope for rebuilding, with the departure of senior squad members, such as Nemanja Vidic, imminent. The perceived failure of several others this season would likely give him a mandate to make further changes (as Moyes is also anticipating).
Ultimately, Van Gaal would have to decide the kind of environment in which he is looking to work at this stage of his career.
Both Manchester United and Tottenham Hotspur offer tempting challenges for a manager who has experienced just about everything at the highest level. For a little less stress, though, Van Gaal would be wise to pick White Hart Lane.