Sky sources understand United are prepared to match the £28million buyout clause in the midfielder’s contract.
United are aiming to get the deal - which would see Herrera sign a four-year contract - done by the end of this month as the buyout clause raises to £32m after 1 July.
Herrera was the lynchpin of Bilbao's midfield last season as they finished in an impressive fourth place in La Liga.
He scored five goals in the campaign, but despite being one of the most impressive performers in domestic football, he is yet to break in to the Spain team.
David Moyes was reportedly targeting Herrera, 24, last summer, but failed to land the central midfielder. There were bizarre scenes in Spain when, on transfer deadline, three men claiming to be representing United tried to negotiate the deal at the Spanish Football Association headquarters. The negotiations broke down and it transpired that United knew nothing about the three men.
But Moyes' successor as Manchester United boss, Louis van Gaal, looks set to pull off the deal as he begins rebuilding the squad.
Herrera was born in Bilbao but came through the ranks at Real Zaragoza before joining his hometown club in 2011 on a five-year contract.
Sky Sports News reporter James Cooper said: "It is very, very close. It is a question of when, not if. It is more likely to be on Wednesday but Manchester United are very close to making their first signing of the summer and the first signing under Louis van Gaal.
"But Manchester United are proving they can go and get a player. It's a question of 'if at first you don't succeed, try, try, try again. Last summer there were some farcical scenes. Manchester United didn't match the buy-out clause. We saw the three stooges arrive at the Spanish FA headquarters - and the whole thing tumbled from there.
"United have their own lawyers on the case this time and they have met the buy-out clause. Now it's a medical and personal terms with the player, and the beginning of a new era at Manchester United.
"A £30m fee is a big fee perhaps for a player who is 24 years old but who has not played for the Spanish national team. He did play for the Olympic team in London in 2012. But Manchester United have started their transfer business - and there will be plenty more, I promise you that."
Southampton and England left-back Luke Shaw could be the next target - United have reportedly upped their offer for him to around £34m.