The 35-year-old Northern Irishman succeeds Paul Cook, who was sacked by the Sky Bet League One club last week.
McKenna was part of Ole Gunnar Solskjaer’s coaching team at Old Trafford and stayed on after the Norwegian was sacked in November.
Martyn Pert, another member of the United coaching set-up, will be his assistant at Portman Road.
McKenna told the club’s website, www.itfc.co.uk: “I’d like to thank (chief executive) Mark Ashton and the owners for putting their faith in me and Martyn to take the club forward.
“Leaving a club like Manchester United was obviously a difficult decision, but I strongly believe in the opportunity to build something here.
“It feels like the right time, project and club to make my first step into first-team management.
“For now, the focus for the team has to be on a huge game on Saturday. After that, I cannot wait to meet everyone and get to work.”
McKenna played for Tottenham’s academy before a hip injury ended his career early. He took up a youth coaching role at Spurs before moving to Old Trafford to work with the under-18s. He was promoted to the first team by Jose Mourinho in 2018.
He takes over an Ipswich team that are 12th in League One and were dumped out of the FA Cup by League Two Barrow on Wednesday. They host third-placed Sunderland at the weekend.
They lost two and drew one of their three games under caretaker boss John McGreal.