Mourinho has made no secret of his displeasure with the defensive options available to him at Old Trafford, having seen the club miss out on a number of centre-back targets during the summer transfer window.
United have subsequently endured a tumultuous start to the season both on and off the field, leading to intense speculation over Mourinho's future.
The club's hierarchy have dismissed reports that the Portuguese boss is on the verge of being sacked, but ESPN FC reports that Mourinho wants to be backed both in public and in the transfer market.
Inter Milan's Skriniar and AC Milan's Romagnoli are both valued at around £40m, and the United boss is understood to have asked chief executive Ed Woodward to sign the 23-year-old duo when the transfer window reopens in January.
Both players would be eligible to play in the Champions League knockout stages should United make it that far, and Mourinho is hopeful that their age will convince Woodward to invest the money he refused to during the summer.