Erik ten Hag's side completed the signing of Eriksen on a three-year contract earlier in the month as he signed a deal until 2025 following his release from Brentford at the end of last season.
Eriksen will inherit the No.14 shirt that was vacated by fellow midfielder Jesse Lingard when he left the club on a free transfer. Lingard has since signed for Premier League new boys Nottingham Forest.
The 30-year-old scored on his unofficial first outing for United on Wednesday, netting a free-kick as his new side defeated Wrexham 4-1 in a behind-closed-doors friendly match.
The Denmark international and fellow summer signing Lisandro Martinez both played 45 minutes of the match at Carrington before being replaced at half-time with the scores level at 1-1.
Eriksen and Martinez could both make their first public appearances this weekend when United face Atletico Madrid and Rayo Vallecano in a double-header.
Ten Hag encountered Eriksen at Ajax at the start of the year when Eriksen trained with his former club to build up match fitness before signing for Brentford.
Upon signing for United, the former Tottenham star said: Manchester United is a special club, and I cannot wait to get started. I have had the privilege of playing at Old Trafford many times but to do it in the red shirt of United will be an amazing feeling.
I have seen Eriks work at Ajax and know the level of detail and preparation that he and his staff put into every day. It is clear that he is a fantastic coach.
“Having spoken with him and learned more about his vision and the way he wants the team to play, I am even more excited for the future.
“I still have major ambitions in the game, there is a huge amount that I know I can achieve, and this is the perfect place to continue my journey.