The Daily Mail says United approached Villa about a loan arrangement for their Argentina goalkeeper.
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United moved for Martinez with Andre Onana set to miss the start of the season due to a hamstring injury.
However, after making their approach over the weekend, United's enquiry about a loan arrangement was rejected by Villa with "incredulity".
Martinez is on a deal at Villa to 2029 and it's suggested Villa saw no positives in sending him to Old Trafford in a temporary deal.