Loretta, 37, hit .280 in 261 at-bats for the Astros last season, including a .330 mark against left-handers.
The D-backs are searching for a second baseman to replace Orlando Hudson, who is expected to sign elsewhere. One baseball official said the D-backs made a "strong offer" to Loretta and that the two sides could be close.
"Some deals are closer than others," Byrnes said speaking in general terms. "We're not quite there yet, at least in our mind."
There had been speculation in some quarters that the choice by reliever Brandon Lyon not to accept salary arbitration, in addition to the club's decision not to offer it to outfielder Adam Dunn, might free up some money that the D-backs would then use to sweeten their offer to left-handed pitcher Randy Johnson.
Barry Meister, who represents Johnson along with Alan Nero, said there had been no new contact since negotiations broke off last month, and Byrnes echoed that.
"Probably not a lot has really changed since they filed, at least from our position," Byrnes said. "He's had an opportunity to assess the market [but] probably not a lot has changed at this point."
While second base is the priority, the D-backs would also like to add some depth in the bullpen, particularly with a left-hander. They have made contact with the lefties on the market, such as Arthur Rhodes, Alan Embree and Joe Beimel, but appear to be willing to let the market come down a bit before plunging in.
The D-backs have three righties that they like late in games in Chad Qualls, Tony Pena and Jon Rauch.
"I think it's been fairly well documented we are looking at some lefties, and if we do get one, we'll have three good late-game righties to go with a lefty," Byrnes said.
Embree, who had his $3 million option declined by the A's, played for the D-backs in 1998. He is likely too expensive at this point, but that could change if the market does.
"I like the area, I like the organization, I liked my time there," Embree told MLB.com recently. "It's an enjoyable place to play. And they were in the [National League Championship Series] two years ago, so obviously it's an attractive place to play in that respect, too. Anywhere on the West Coast or in the West somewhere would be something I'd be interested in."
Another reliever the D-backs may have an interest in is former Nationals closer Chad Cordero. A source said that Cordero was due to arrive in Las Vegas on Monday night and that he was planning on meeting with the D-backs, Mets, Cardinals, Rangers and Tigers.
A published report said the D-backs had decided to trade catcher Chris Snyder instead of Miguel Montero, but multiple sources said that the club had no intention of dealing Snyder.
Arizona, however, is willing to part with Montero, who at this point appears to be their lone trading chip. The Red Sox, who have a surplus of young pitching could be a match. While they would not deal someone like Clay Buchholz, they might be willing to part with someone like Michael Bowden.
The D-backs also signed four Minor League free agents. Outfielder Trent Oeltjen, infielder/outfielder Ryan Roberts, infielder Josh Wilson and right-handed pitcher Seth Etherton.
All four also received invitations to big league camp during Spring Training.