According to Vazquez's agent, Melvin Roman, the D-backs last week offered a one-year deal plus a club option for 2010.
"It was for less than what another club is offering to us," Roman said. "[The D-backs] said they would get back to us with something else, and that's what we're waiting on right now."
Vazquez, 32, has been one of the club's top targets after he hit .290 with a .365 on-base percentage for the Rangers last season. His OPS+ (a weighted measure of on-base percentage plus slugging) was 111 last year, which means according to that statistic he was 11 percent better than league average at the plate.
Vazquez told Texas media in September that he wanted a multiyear contract and that is still his desire.
"That would be a priority for us," Roman said.
The D-backs are looking to replace free-agent second baseman Orlando Hudson and would apparently like to do it with a combination of Vazquez and a right-handed hitter such as Mark Loretta or Damion Easley.
Steve Gilbert is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.