Both players collected Portuguese winners' medals for the third season in a row earlier this month and a number of European clubs were believed to have been chasing their signatures before Russian billionaire owner Dmitry Rybolovlev opened his chequebook.
Portuguese international Moutinho, who has spent the last three years with Porto since joining from Sporting Lisbon in 2010, came at a price of €25million.
News of the 26-year-old's transfer will be a major blow for Tottenham Hotspur, who had been expected to revive their interest in the midfielder this summer after missing out last year.
Colombian winger Rodriguez has collected eight trophies in a three-year spell in Portugal and has commanded a €45million fee.
A statement issued by Porto read: "Futebol Clube do Porto would like to inform the market that it has reached an agreement with Association Sportive de Monaco Football Club for the permanent transfer of players James Rodriguez and Joao Moutinho.
"The combined fee of this transfer is €70million (£60m), €45m (£38.4m) for the transfer of James Rodriguez and €25m (£21.4m) for the transfer of Joao Moutinho."
Claudio Ranieri's side have also been linked with a big-money move for Atletico Madrid striker Radamel Falcao, who has attracted interest from Real Madrid, Chelsea and Manchester City.