The team scheduled a 12:30 p.m. PT news conference for Thursday to announce the deal. Aybar is also scheduled to receive his 2011 American League Gold Glove award Thursday before the Angels' game against Oakland in Anaheim.
Aybar's contract will be worth a guaranteed $35 million over four years, paying him an average annual salary of $8.5 million, sources told Enrique Rojas of ESPNdeportesLosAngeles.com.
"I don't think we ever thought it wouldn't get to that point," Dipoto said earlier Wednesday. "It took a while. There are just some details to sew up but we are very close."
Aybar, 28, is a .274 hitter in parts of seven seasons with the Angels. He's batting .171 (7 for 41) through the team's first 10 games this season.
Aybar would have been eligible for free agency this offseason.
Aybar said Wednesday that his agents pointed out that there could be shortstop vacancies on the New York Mets, Boston Red Sox and Los Angeles Dodgers this winter and that he could have been better served to wait for the free agency period.
Like pitcher Jered Weaver did last August, he went against his agent's advice to stay with the team that signed him out of the Dominican Republic in 2002.
"I'm doing it because I really like this team. They've given me a lot of support, even when I was nobody," Aybar said.
Once the Angels announce Aybar's deal, they officially will have locked up their double-play combination for the long-term -- second baseman Howie Kendrick agreed to a four-year, $33.5 million contract in January.
Information from ESPNLosAngeles.com's Mark Saxon and ESPN.com's Jerry Crasnick contributed to this report.