Morsy has agreed a deal to drop from the Championship and link up again with his former boss, Paul Cook.
The Blades had been linked with the Egyptian international midfielder after Boro expressed an interest in signing forward Burke.
But Morsy, who captained both Chesterfield and Wigan under Cook, became Ipswich’s 19th signing of a busy summer ahead of tonight’s 11pm deadline.
“I’m absolutely delighted,” Cook said.
"Samy, again, has been a long-term target.
"I can only thank Mark Ashton and the owners for their continued backing for bringing in a strong character and a good player like Samy Morsy."