Delap, 22, scored 12 Premier League goals during his sole season at Portman Road but could not prevent Ipswich making an immediate return to the Championship.
Manchester United and Everton were also reportedly willing to meet the buyout clause in the England under-21 international’s contract.
But Chelsea have won the race for the former Manchester City forward, bolstering their attacking options ahead of the Club World Cup in the United States, starting later this month.
“I understand the stature of this club and can see the trajectory it is on with these players and the head coach,” said Delap, who has signed a six-year contract, in a Chelsea statement. “It’s going to be an incredible place for me to develop, and I hope to achieve amazing things here and help the club win more trophies.”
The Blues finished fourth in the Premier League and will return to the Champions League next season after a two-year absence.
And Enzo Maresca’s men last week beat Real Betis 4-1 in the Conference League final to secure a trophy in the Italian manager’s first season in charge.
In mid-December Chelsea were within touching distance of eventual Premier League champions Liverpool before hitting a rocky patch.
They finished the season strongly, securing a top-five spot on the final day of the season but were hamstrung by their lack of a clinical number nine.
Senegal international Nicolas Jackson scored just 10 times in 30 Premier League appearances, while Christopher Nkunku looks set to leave the club after an underwhelming two-year spell.
Delap, the son of former Stoke midfielder Rory Delap, began his career at Manchester City but struggled to break into Pep Guardiola’s star-studded first team.
After loan spells in the second tier with Stoke, Preston and Hull, he was snapped up by Ipswich after they won promotion to the Premier League a year ago.
Delap was one of the few success stories for the Tractor Boys during a difficult season as they were relegated back to the Championship with just 22 points.
Chelsea will face Flamengo, LAFC and ES Tunis in the group stages of the Club World Cup, which gets underway on June 14. (AFP)