Following their 6-2 thrashing of Leeds at Old Trafford, United moved five points behind leaders Liverpool, having played one game fewer.
Ole Gunnar Solskjaer’s side are unbeaten since November 1 and have won six out of their last seven league games.
And Shaw believes that with their extra consistency, they can mount a serious title challenge.
“The way the Premier League is going at the moment, I think it would be silly to doubt us,” Shaw told Sky Sports. “I think at this moment in time, it’s anyone’s really.
“The team that takes charge, goes well over the Christmas period, and wins all their games, I think have got a very good chance.
“We just need to try and get consistency in our game, because we can’t keep having one good game and dropping off and then having one bad game.
“There’s no way you can mount a title challenge like that so I think we need to keep consistency and see where it takes us.
“The Champions League really hurt, but I think performance-wise if we get that consistency, then we’ll be in a much better place than we are already.
“Of course, we’ve got lots to improve on, all teams do at times, but if we can keep improving and keep that consistency, I’m sure we’ll be in a much better place.”
Despite moving themselves into the title picture in recent weeks, Solskjaer refused to get carried away, claiming he won’t look at the league table until spring.
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“We are not even at the halfway point,” he said, following the win over Leeds. “The league position is not something we look at – we look at the improvement of the team. Now we are getting fitter and stronger. Our fans needed that one.
“I’m confident myself and my staff have enough experience Mick Phelan had the best seat learning from Sir Alex. Me and Michael experienced it as players. We won the league quite a few times.
“Of course it is different on the staff. Our job is to make sure the players focus on the next game and if we get to March and are in that position, we are going to enjoy it.”
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