Dombrowski, speaking at a Boston baseball writers event, said: "I still don't expect us to have a high expenditure on our closer situation."
Last year's closer, Craig Kimbrel, is a free agent seeking a long-term, high-salaried contract, but Dombrowski noted concern about the payroll luxury tax.
"We have some flexibility, we don't have a lot," he said.
The Yankees added Adam Ottavino on Thursday, ESPN's Jeff Passan reported, giving Boston's AL East rivals a bullpen that already has Aroldis Chapman, Dellin Betances and Zach Britton, all among the best set-up men in baseball.
"We do feel we have guys who can get outs in the ninth inning, and in the seventh and eighth," manager Alex Cora said. "(The Yankees) have a great bullpen, but we have a great rotation. It balances out."
Dombrowski echoed that, saying: "When you look at our starting rotation, I think it's second to none."
Cora also hedged on the team's visit to the White House, which has been scheduled for Feb. 15. He said "right now" he's committed to going, "but that might change tomorrow."
Cora has been critical in the past of some things President Trump has said about the manager's native Puerto Rico.
"If I go, I'll represent Puerto Rico the right way," Cora said. "I don't know what kind of platform I'll have if I go. It's not that I've changed my mind, but we'll see what happens in the upcoming days."
Holt made note of the fast-food spread Clemson received when the college football national champions visited the White House on Monday and said, "'If that's the case, I hope we get Chick-fil-A or something.'