The 32-year-old is currently a free agent after terminating his contract with Atletico Madrid, and he has already lifted the Premier League title twice with Chelsea in 2015 and 2017.
Richards - a title-winner himself with Man City in 2012 and 2014 - believes that Costa could be the focal point that Guardiola's attack needs with Sergio Aguero out injured.
"Manchester City are already on a ridiculously good run of form but if they signed Diego Costa now, he would win them the Premier League - I am sure of it," Richards wrote for BBC Sport.
"Why not get him in until the end of the season? He would do more than just ruffle a few feathers and be a pain in the backside for opposition defences - he is a proven finisher who would offer Pep Guardiola's side a proper plan B.
"Yes, you would have to cover Costa's wages, but the massive difference he could make to City's season would make that worthwhile.
"I know Pep said last week that City are not signing anyone in this window, and that stance might be because of Financial Fair Play rules - City could be saving their money to go for one of the big boys in the summer, like Borussia Dortmund's Erling Braut Haaland or Paris St-Germain's Kylian Mbappe, who are not available or affordable now.
"Of course those two would be top of anyone's list, but that kind of transfer can wait. I am talking about who they can bring in now, just for the next few months to give them that extra bit of firepower to get over the line, and Costa fits the bill."
Costa scored twice in seven appearances for Atletico in the first half of the season and was previously linked with a move to Wolverhampton Wanderers, who have since signed Willian Jose on loan from Real Sociedad. body check tags ::