The Sun newspaper says United boss Jose Mourinho has instructed the Old Trafford hierarchy to pursue a move for the defensive duo, who signed new five-year deals to stay at White Hart Lane in September.
City have reportedly been monitoring Walker and Rose ever since Pep Guardiola's move to the Etihad was first discussed two years ago, with the former Barcelona boss understood to be an admirer of Rose in particular.
Both players do not have release clauses in their contracts so a double-deal could cost in excess of £60m ($73m).
Guardiola has made signing defensive reinforcements a priority and has been linked with a January move for Bayern Munich's Holger Badstuber.
"We are evaluating if there is a chance, a possibility, to take one. I know Holger, he's a fantastic player, a fantastic guy, but he's a Bayern Munich player," the Spaniard was quoted as saying by the Daily Mail after City's FA Cup win over West Ham.
"Of course first of all we should speak with them, with him. Now, on the table, there are no names, because there were a lot of games. Now we have 10 days off, until Everton game."
Meanwhile, Mourinho expressed displeasure with the performances of Luke Shaw earlier this season and has deployed midfielder Antonio Valencia as a makeshift right-back at Old Trafford.
The Portuguese coach said in December that he would not make multiple signings in the January transfer window.
"I believe more in the summer market than the winter transfer window as it gives you more stability, more time and more choice," he was reported as saying by ESPN.
"Let's see what happens. If we buy a player we buy a player, but we are not buying two, three or four."