Rojo's move to Boca Juniors is nearing completion but there is no rush to finalise the deal before Monday's deadline as Argentina's transfer window open until 19 February.
The 30-year-old has not appeared for United since November 2019 and spent the second half of last season on loan at Estudiantes.
Rojo travelled back to Argentina over Christmas and was granted permission to speak to other clubs in the hope of securing a move before his contract expires at the end of the season.
After signing a new three-year deal in March 2018, Rojo made just 17 appearances for United and only 12 under manager Ole Gunnar Solskjaer.
Mengi, a highly-rated centre-back and captain of United's Under-23s, is set to join Derby County on loan to become Wayne Rooney's first signing as a manager.
The Championship club have been unable to sign players this month due to a transfer embargo after failing to pay their players' December wages in full.
Mengi will become the latest United player to leave on loan after Jesse Lingard and Facundo Pellistri agreed to spend the rest of the season at West Ham and Alaves respectively.
Tahith Chong and James Garner saw their loans at Werder Bremen and Watford over the weekend, with the pair joining Club Brugge and Nottingham Forest on new temporary deals respectively.
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies