Rayo Vallecano accepted a £2m bid for the 26-year-old and the Premier League club will seal a third summer signing, subject to international clearance.
Michu has passed a medical and agreed personal terms to become new Swansea boss Michael Laudrup's third signing.
Laudrup said: "We're talking about a player who scored 16 goals in a small team in the Spanish league."
Michu said: "I really wanted to come to Swansea because of the coach, the style of play and the Premier League. I had other offers.
"I've always followed the Premier League from Spain and I watched a lot of Swansea games last season. They played good football and I liked their style."
He will join fellow new signings defender Chico Flores and midfielder Jonathan de Guzman on Swansea's three-match pre-season tour to the United States on Sunday following Saturday's friendly at Welsh Premier Port Talbot Town.
Unlike De Guzman and Flores, Michu has not previously worked with Laudrup but he said: "I also know how Michael Laudrup's teams play, so I'm looking forward to being part of that.
"The coach was fundamental to me joining Swansea. I remember watching him as a player - he was a machine."
The former Celta Vigo playmaker scored 15 goals in 37 games in his only season for Vallecano in La Liga last term.
Vallecano's La Liga rivals Sevilla were also believed to be interested in signing Michu while he had also been linked with Swansea's Premier League rivals Manchester United, Fulham, Stoke City, Liverpool, Southampton, Wigan and West Bromwich Albion.
Michu, an attacking midfielder, is seen as an ideal replacement for Gylfi Sigurdsson who, after an impressive loan spell at the Liberty Stadium last season, has joined Tottenham Hotspur from Hoffenheim.
"I am happy that the player wanted to come here because I was here," said Laudrup. "I have never worked with him and I haven't paid him!
"He has had a press conference in Madrid to say goodbye to Rayo Vallecano and he should join us for the tour to the USA."
Swansea City have completed the deal to sign Michu as the Spanish midfielder has agreed a three-year contract.