The 23-year-old has been one of the standout players for the Basque club and has long been linked with a move to the Etihad. City manager Pep Guardiola has been an admirer of the central defender.
Laporte had an opportunity to join the Premier League club, but turned it down when the Catalan coach arrived at the club in 2016. His current contract at Bilbao runs down in 2020 and has a €65m (£57.1m, $80.1m) release clause in his contract.
City already lost Alexis Sanchez to Jose Mourinho's side after the Chilean international snubbed the league leaders for United.
Neville managed Valencia for four months between December 2015 and March 2016, during which his side faced Bilbao on three occasions. He is aware of Laporte's qualities and now wants United to beat City for the second time by completing the defender's signing in the January transfer window.
"In my very short time in Spain we played them 3 times ! This lad can play! Get in United. Yes. Its really hard to get players out of Bilbao and costly but this lad is a very good player," Neville tweeted.
According to the Manchester Evening News, Laporte was convinced that Guardiola and City would make another approach for his services, despite rejecting the Premier League leaders 18 months ago.
The former Barcelona and Bayern Munich manager has the services of John Stones, Nicolas Otamendi, Vincent Kompany and Eliaquim Mangala as the four recognised centre-backs in the squad.
Kompany has struggled with injuries over the last few years, while Mangala has not been preferred by Guardiola. City are keen on strengthening their defence for the second half of the season and Bilbao star has re-emerged as a transfer target in January.
Laporte rejected City's advances in Guardiola's first summer transfer window because he wanted to further develop as a player. He believes the time is right for him to leave Bilbao and wants to take the opportunity to swap San Mames for the Etihad in January. He is also hopeful that a switch to England will improve his chances of making it to France squad for the summer World Cup in Russia.