United are understood to have had two bids of 40million euros (£27.8million) and 45million euros (£31.3million) rejected by Real Madrid, who value the long-serving defender at a staggering 90million euros (£62.7million).
With Real’s interest in United keeper David de Gea further complicating matters, the two teams have both stood firm in their individual valuations of their players so far.
Now, Spanish daily AS claims United are to make a final approach for Ramos of £42million as Louis van Gaal bids to draw a line, one way or another, under the saga.
Ramos has been at loggerheads with Real all summer over a new deal and is widely reported to be keen to make the move to Old Trafford.
The 29-year-old is said to be requesting a wage of £7million a year, however, the La Liga side are reluctant to meet his terms.
However, United are expected to offer the Spain international an even better deal, putting up a fee of £7.6m – similar to club captain Wayne Rooney.
Should Real reject United’s final bid for the player, Van Gaal may have to turn his attention elsewhere in his bid to strengthen his defence, with an offer for another long-term target Nicolas Ottamendi, looking increasingly likely.