Sky sources understand Sherwood is set to be relieved of his duties at the end of the season with Spurs eight points off fourth place and 50/1 to qualify for the UEFA Champions League.
Van Gaal, who is set to leave his role as Holland manager after this summer's World Cup, was installed as the 1/2 favourite to replace Sherwood and has shortened slightly to 2/5.
Southampton boss Mauricio Pochettino (10/1), Ajax head coach Frank de Boer (14/1) and Spurs legend Jurgen Klinsmann (16/1) are considered the most likely alternatives.
Sky Bet's head of football trading Paul Wiggins said: "We installed van Gaal as the clear favourite upon opening the market with suggestions that Spurs chairman Daniel Levy had already been in talks with him.
"Our punters certainly seem to agree with plenty backing him ahead of other candidates despite the short odds, forcing us to cut the Dutchman's price further."
Van Gaal is also currently proving the most popular option in the skysports.com poll on who will replace Sherwood this summer.