The Uruguay international is in the final year of his contract, and Woodward "has held several rounds of positive talks with Atletico," according to Jeremy Cross of the Daily Star.
Woodward is said to be confident of securing the defender's signature, and he is prepared to pay his £20 million buyout clause.
United neglected to improve their options in central defence over the summer.
It looks to have been a costly decision, as the Red Devils are eighth after 12 games with a negative goal difference. United have conceded 21 goals, the same as Southampton in 17th place and better only than the records of Burnley, Cardiff City, Huddersfield Town and Fulham.
Meanwhile, Godin has been part of a defence that conceded just eight goals in La Liga this season and last term only conceded 22 times in the Spanish top flight. In the 2015-16 season, Godin helped Atleti ship just 18 goals.
Now in his ninth season with Los Rojiblancos, the centre-back is one of the best in the world at his job.
An uncompromising leader at the back, Godin commands Atleti's defence, reads the game well and is equally combative on the ground or in the air.
Tom Allnutt of AFP shared a recent example of Godin's commitment to the cause, as the defender helped Atletico come back to beat Athletic Bilbao 3-2:
The 32-year-old has previously played in two UEFA Champions League finals. He even scored in the first of those, but on both occasions, his side were defeated by local rivals Real Madrid.
Despite his age—he will be 33 in February—he remains an outstanding player, so he could be a worthwhile purchase for United, even if he would only be a stopgap.
He was linked with a move to United in the summer, and ESPN's David Cartlidge would have been surprised if he had joined the Manchester outfit:
The same could be the case in January, though if Atletico opt against giving him a new contract, they may be happy to let him go so they can bring in a fee for him.
United's interest might also spur them on to tie him down on a new deal. If that ends up being the case, it's imperative the Red Devils look elsewhere. They are in desperate need of some improvement at the back.