The Dutchman endured a tricky start to his reign but seven wins in eight league games have put United hot on the heels of the pacesetters.
If the current United side can emulate the teams of Alex Ferguson who were renowned for a strong second half of the season, Van Gaal's men could yet have their say in the destiny of the title.
United visit Tottenham Hotspur in a lunchtime kickoff on Sunday as the games continue to flow thick and fast over the festive period.
"The signs have been good in recent weeks," midfielder Michael Carrick told the club website (ManUtd.com).
"We were bitterly disappointed not to win last week (at Aston Villa) but we beat Newcastle and now have two big away games. They will be a good test for us to see where we are at."
After the trip to Spurs, United -- seven points behind City and 10 adrift of Chelsea -- are away at Stoke on New Year's Day.
"We’re continuing to look up and will continue to try to catch them (City and Chelsea)," Carrick said.
"We’re not satisfied with third -- it’s obviously an improvement and we’re getting better, but we want to keep looking up."
Seventh-placed Spurs have won three on the bounce with striker Harry Kane emerging as one of the stars of the season, his first-minute strike in the 2-1 victory over Leicester on Friday taking his tally to 15 in all competitions.
"We've built up some momentum now with some wins," he said.
"United have picked up form. It's going to be a tough game but we are confident at the minute. We're ready for it."
Chelsea make the trip south to face a fourth-placed Southampton side who appear to have got over their blip following a flying start to the campaign.
"We have found the belief again," said Southampton manager Ronald Koeman after Saints followed up a 3-0 drubbing of Everton with a 3-1 success at Crystal Palace on Friday.
"We know Chelsea are strong. It will be a challenge."
Manuel Pellegrini's City will set a club record of 10 wins in succession in all competitions if they can take three points at home to Burnley.
(Editing by Ed Osmond)
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