The Argentina international was transfer-listed at the end of last season and the two clubs have been haggling over a price since then.
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United have refused to budge on their initial £50m asking price and have even rejected a swap offer by Chelsea involving Tyrique George.
For their part, Chelsea had viewed Garnacho as being rated at closer to £30-35m.
However, the Daily Mail says Chelsea are hoping for a breakthrough after their latest offer of £35m plus bonuses.
For his part, Garnacho is determined to see the move happen, having settled on personal terms with Chelsea some weeks ago.
