Milner is out of contract in the summer and, throughout the season, has held off signing a new deal that was offered last summer.
That has alerted other clubs - and Arsenal and Liverpool have been heavily linked - but Press Association Sport understands City have now raised the stakes by tabling an enhanced deal.
Reports have suggested this could be worth as much as £165,000 over four years, making him one of the club's top earners.
Yet it still remains to be seen whether Milner, 29, will commit as he ponders what could be the last big move or contract of his career.
Manager Manuel Pellegrini has said that money and terms have not been an issue for Milner, but that the matter comes down to playing time.
Milner is thought to want to be sure that he is likely to play a prominent role at the Etihad Stadium having been frustrated by a lack of starts in last season's title-winning campaign.
This season he has been more active, making 27 starts in all competitions, including some of the biggest games of the campaign, despite needing to manage a knee condition.
Pellegrini has repeatedly said Milner is a valued member of his squad, and his versatility has been an asset, playing in central and wide midfield roles, as well as up front.
Asked about Milner at the weekend, Pellegrini said: "It's a decision James must take at the end of season, what is better for his future.
"I always talk that he is an important player for our team, he can play in different positions because he is an intelligent player.
"He always gives 100 per cent but it will be sorted at the end of the season."
Milner joined City for £26million from Aston Villa in 2010 and has previously played for Leeds and Newcastle.
Keeping Milner would also help City with regards to their homegrown quota for next season.
One English player poised to leave is winger Scott Sinclair, whose loan to Villa is set to become a £2.5million permanent deal this summer.
It is understood this was agreed when Sinclair moved to Villa Park in January and would only not occur if the midlands club were relegated
This will now not happen with Villa's top-flight status having been secured at the weekend.
Doubt was raised because Villa manager Tim Sherwood said recently he had not seen confirmation Sinclair would be signed permanently, but the deal was in place before he took charge.
Sinclair has scored three goals for Villa but not started since March as the FA Cup finalists have switched from playing wide men to a different system.
Despite that, Sherwood did say the 26-year-old could have a role to play.
He said: "I think if we go back to wide players then Scott will be an automatic choice to play there.
"It's horses for courses at the moment
That's the way we're choosing to play.
"He hasn't become a bad player or anything like that."
Source : PA
Source: PA