Delap spent the last season on loan at English Championship side Hull City, scoring eight goals in 32 appearances.
Neither Ipswich nor City disclosed the financial details of the signing, but British media reported that the deal was worth about 15 million pounds plus add-ons.
Delap, who is the son of former Stoke City midfielder Rory Delap, has represented England at various youth levels and becomes Ipswich's fourth signing of the transfer window. He has also had loan spells at Stoke and Preston North End.
"He is a player with the qualities to help the team this season but also to be an important player for the club for many years to come," manager Kieran McKenna said in a statement, praising his "outstanding physical and technical attributes."
"He has gained good first-team experience from a young age and we feel he has the potential for development here," McKenna added.