Pep Guardiola is desperate for new full-backs and needs young English players to boost his homegrown quota but City are baulking at Rose’s valuation.
It adds to the uncertainty around Shaw whose Old Trafford future is in the balance after repeated public criticisms from Jose Mourinho.
But Shaw, 21, signed from Southampton in June 2014 for £27million, insists he wants to win over the United manager.
He held talks with Mourinho on Monday before he came off the bench in the 65th minute of Tuesday’s 1-1 draw with Everton.
Mourinho praised his display, which saw Shaw earn a late penalty to save a point when Ashley Williams handled his shot.
But Mourinho still says he wants Shaw to play with more maturity on the pitch and Mirror Sport understands he also wants changes off the field too.
Shaw currently lives in the North West with three old friends from Molesey, Kingston upon Thames, where he grew up.
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Phil Neville thinks Jose Mourinho is giving Luke Shaw his last chance
The trio moved up to keep Shaw company after he landed his big-money move.
But the student-style arrangement is not ideal for a professional sportsman and both Mourinho and Shaw’s family are now demanding a change.
The England international, who has seven caps and earns £80,000-a-week, is under pressure to be more dedicated.
Shaw is not alone in feeling Mourinho’s wrath as other youngsters including Jesse Lingard have also frustrated him with their lifestyles.
But it is Shaw’s decline in form and prospects which has been most staggering.
A move to link up again with his former Southampton mentor Pochettino could help salvage his career.
But a switch to Tottenham is unlikely unless Rose, who signed a new five-year deal in September, is sold.
Liverpool and Man City are also keen on Shaw and need new left-backs but United are reluctant to do business with local rivals.
It leaves Shaw at a crossroads facing an uncertain future.
He hopes Mourinho’s criticisms are ‘tough love’ to fire him into improvements but cannot be sure his decision to snub Mourinho and Chelsea in 2014 is not a factor.
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Mourinho says HE was making every decision for Luke Shaw after effective performance
Shaw has endured a difficult time since ex-boss Louis van Gaal questioned his fitness when he first arrived from Saints, and then he suffered a badly broken leg in September 2015.
Mourinho also publicly criticised Shaw earlier this season for ruling himself out of their November match at Swansea with injury.
He then claimed earlier this year that the starlet was his FOURTH choice left-back, behind Daley Blind, Marcos Rojo and Matteo Darmian.
But Shaw’s return on Tuesday at least gives him a chance of winning over Mourinho despite repeated criticisms from his boss – including for talking to Tyrone Mings in the tunnel at half-time of the controversial Bournemouth game.