Reports are that the Rangers made what is believed to be a four-year, $48-million proposal.
"Tom Hicks said that he was going to come to Puerto Rico and give it his best shot," Delgado's agent, David Sloane, said in an e-mail. "He did all that and more."
Texas, competing with the New York Mets, Florida and Baltimore, made progress during the four-hour session in Puerto Rico.
"I think they recognize it was a very strong offer, I think they recognize we went certainly farther ... it was more aggressive than anything we talked about before," Hicks said. "I think we made a very compelling offer."
Hicks wouldn't go into details about the offer. He expects Delgado will decide on the Rangers' offer by the end of the weekend.
Sloane said he expected to talk to Texas again this weekend "in an attempt to sustain the positive momentum we generated today."
General manager John Hart and manager Buck Showalter, who got contract extensions Friday, joined Hicks for the meeting. The group then returned to Texas. Hart and Showalter had also met with Delgado during the winter meetings last month.
New York Mets chief operating officer Jeff Wilpon and general manager Omar Minaya met with Delgado in Puerto Rico for about three hours Thursday. It was Delgado's second meeting with the Mets.
Delgado would be an impressive addition to a youthful Texas lineup that includes All-Stars Alfonso Soriano, Michael Young and Hank Blalock and switch-hitting slugger Mark Teixeira, like Delgado a first baseman.
Hart said there was "a lot of non-money conversation" with Delgado, including the dynamics of the lineup and said that issue wasn't a sticking point. There was no commitment made to Delgado about how much he would play first base or be the designated hitter.
But Teixeira went to Hart's office unsolicited in October and said he was willing to change positions if there was the chance to get a player like Delgado.
Delgado hit at least 30 homers in each of the past eight seasons in Toronto. In 155 career at-bats in Arlington, he has hit .316 with 19 homers and 43 RBI.
Information from The Associated Press was used in this report.