The Red Devils were close to sealing a deal for Sancho last summer and had agreed on terms with the Dortmund forward.
But Man United failed to meet Dortmund’s demands of £108million by a self-imposed August deadline, and the deal fell through.
United are expected to rekindle their interest in the England international this summer but may have competition from elsewhere.
Klopp’s Liverpool are in need of new attacking recruits, with Mohamed Salah, Roberto Firmino and Sadio Mane all 28 and above.
And Borini has suggested Liverpool beat United to the punch and sign Sancho ahead of their Premier League rivals.
“Looking at Liverpool’s competitors I think more depth in the attacking positions would do the team good,” Borini told CaughtOffside.
“Jadon Sancho would be a great signing and it would be fun to steal him from Man United.”
Liverpool are expected to undergo something of an overhaul in the summer with Mane and Salah both linked with the exit.
Netherlands international Georginio Wijnaldum is also expected to leave for Barcelona unless Liverpool can tie him down to a new contract between now and June.
The Reds are currently only linked with a centre-back though, RB Leipzig star Ibrahim Konate.