Woodward made the decision to sack Jose Mourinho last month after a dismal start to the season.
Ole Gunnar Solskjaer replaced the Special One, with the Red Devils icon winning his first three games in charge.
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Solskjaer was appointed on an interim basis until the end of the season.
United are then expected to pursue Tottenham’s Pochettino in the summer.
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And according to The Sun, Woodward is desperate to land the Argentine at the end of the season.
They say United have a shortlist of names to replace Solskjaer, however, Woodward is adamant the right man for the job is Pochettino.
A lot has been made of Pochettino taking over at Old Trafford this summer, with former striker Teddy Sheringham claiming Tottenham will not stand in his way.
“Pochettino is doing all the right things at Tottenham,” he said.
“People will say he hasn’t won anything but he’s transformed things in north London. He’s got them on the right path.
“He gets players playing to the best of their capabilities and that’s what you want as a manager.
“I don’t think it’s a case of whether Spurs will let him go and I don’t think it’s about money either.
“It’s just about what Pochettino wants to do.
“If United want him and he wants to go then I don’t think Tottenham will stand in his way.
“You can’t turn down the chance to join Manchester United as a player and it must be exactly the same as a manager.”
United are back in action on Wednesday evening against Newcastle in the Premier League (8pm kick off).