The Scotland fullback's deal expires at the end of the season.
Former United scout Mick Brown told Football Insider: "They want Robertson to sign a new contract. (Kostas) Tsimikas has been more involved this season and (Arne) Slot likes to rotate those two, but they still feel he's valuable on and off the pitch.
"We've talked about their priorities when it comes to securing their best players. But Robertson has gone under the radar in that regard.
"There are question marks about (Virgil) Van Dijk, (Trent) Alexander-Arnold and (Mohamed) Salah, now there are questions about Robertson, and all of a sudden they are in serious trouble.
"At worst, they lose all three at the end of the season. So if Robertson leaves next year, the majority of their leaders and big names are gone."