Ten Hag wanted to sign the former West Ham captain before he left for Arsenal. United then signed Mason Mount from Chelsea.
"He (Rice) is a very good player and he would absolutely have fit in here. But it's not that I'm not happy with Mason. This is what you are implying. I am very happy with Mason and he is also a very robust player," said Ten Hag.
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"You see in his profile: he played in his last four seasons - this is the fifth - in the last four seasons he only had one small (injury) at the end of his Chelsea career. Before that, he played almost all the games, three seasons in a row, every third day.
Ten Hag emphasises Mount has been unlucky with his injuries this season.
"It is really unlucky what happened and he is a really good player. We have really missed him, he absolutely could have contributed," he added.