SAN DIEGO -- As the media walked into a suite occupied by Red Sox chief baseball officer Chaim Bloom and general manager Brian O’Halloran at the Winter Meetings on Monday, the executives staged a bit of a prank.

There were three unopened bottles of wine on the conference table and O’Halloran asked the media member at the front of the line to check out the label. The wine was called “Pessimist.”

Bloom and O’Halloran said the bottles were delivered to the room accidentally, but they joked the brand was more fitting for the media to consume, given recent reports suggesting the Red Sox are falling out of the race to re-sign star shortstop Xander Bogaerts.

When the offseason started, Bloom said retaining Bogaerts was the organization’s top priority. He maintained that stance at the General Managers Meetings in Las Vegas last month.

And even as Trea Turner agreed to a whopping 11-year, $300 million deal with the Phillies on Monday that threatened to inflate the market at shortstop, Bloom maintained hope that Bogaerts won’t need change of address forms.

“We are not getting to the point where we’re closing the door on Xander,” said Bloom. “That’s not something we want to envision.”

Turner is only eight months younger than the 30-year-old Bogaerts, and the two players had comparable statistics last season.

If Bloom is worried about how Turner's contract could impact his chances to keep Bogaerts, he isn’t letting on.

“Yeah, it can [have an impact], but we knew this was going to be a robust market and we expected that. It has been,” said Bloom. “Nothing about that has surprised us. This is what free agency is about. Talent is going to get paid.”

What is unclear is how much the Red Sox are willing to pay in terms of average annual value and years to keep Bogaerts, keeping in mind the club needs substantial roster improvement in multiple areas to rebound from a last-place finish in 2022.

Boston is looking to add one to two starting pitchers, and as many as three to four position players, including a starting catcher. Though the Sox recently reached agreements with a lefty (Joely Rodríguez) and a righty (Chris Martin) in the bullpen, they would like to add additional relief options for Cora.

“That involves adding a lot of talent to the group that we had at the end of ’22,” Bloom said. “This is going to be a process that involves adding seven, eight, nine, maybe more players that we had after ‘22 to build a team we want to have that has both impact and depth.

“Some of those moves are going to be smaller, some are going to be larger. The second the offseason starts, you need to be engaged in the whole process. We already are taking steps along the road to build the whole team and what that looks like. And we’re going to continue taking steps along that road, and hopefully Xander is one of those players we add.”

Though the Red Sox tried to strike a deal with Bogaerts when they had exclusive negotiating rights with him throughout the month of October, agent Scott Boras made it clear that his client was going to test the market.

That process is in full motion this week, with Bogaerts meeting directly with teams in San Diego. Bloom doesn’t think it’s necessary for the Red Sox to have a recruiting meeting for someone who has been with the organization since 2009.

A report in The Boston Globe on Sunday, citing a source, said that the Red Sox had yet to make a “competitive” offer to Bogaerts.

“It’s not for me to decide or announce what’s competitive or what isn’t. We’ve certainly made offers to him, and we’ve certainly been engaged and we’re going to stay engaged,” Bloom said.

It could be a situation where Bogaerts gathers all the information he can from other suitors and Boras informs the Red Sox what it will take to keep Bogaerts.

Will Boston get a chance to make a “last, best offer”?

“Usually in any free agency process, there comes a time for that,” Bloom said. “Every [case] is a little bit different. Even within the same offseason, different players, some of it is the players, some of it is the agents, some of it is the club. They all run a little differently. Formal offers, conversation, informal offers, but they come to a head at some point.”