Leeds have not commented on Newcastle's latest offer but it is anticipated that the board at Elland Road will once again reject it.
Newcastle approached Leeds before Christmas with bids of £1m and £1.25m, fees which were turned down outright by United, and a further offer made this week was a last-ditch effort to persuade United to part company their 20-goal striker.
Leeds have no interest in selling Beckford for less than their asking price, thought to be in the region of £2m, despite the fact that the forward's contract will expire at the end of this season.
Newcastle have tested United's resolution repeatedly during the last month but should their latest bid prove unsuccessful, the Championship leaders are likely to take their search for a new forward elsewhere.
Newcastle's board are understood to have set £1.5m as the maximum fee they were willing to pay for Beckford and will not return to Leeds for a fourth time.
Beckford stands to treble his wages by transferring to St James' Park but Simon Grayson warned after the club's FA Cup victory at Old Trafford on Sunday that his club would not bow to a cut-price deal.
Regardless of the size of Newcastle's offer, Leeds would only move to agree a deal once Grayson had lined up a suitable replacement.