Rodriguez was first interviewed by investigators prior to the 2010 season, the Times reported. Rodriguez has never denied having been under the care of Galea but, according to the Times, told investigators that Galea never gave him any performance-enhancing drugs.
The Times reported that recently investigators have tried to get Galea's medical records on Rodriguez but have thus far been unsuccessful.
Galea is currently involved in plea negotiations in Buffalo after being indicted by a federal grand jury in October on five charges of distributing performance-enhancing drugs.
Rodriguez's lawyers issued a statement regarding reports of the ongoing investigation.
"Alex fully cooperated with Major League Baseball and federal authorities in Buffalo regarding his treatment with Dr. Galea, including granting a release of his medical records," the statement said. "Regarding matters before the grand jury, strict secrecy rules do not permit us to comment."
Galea has admitted to using human growth hormone but has insisted that he has never given any of his clients the banned substance. In addition to Rodriguez, Galea has also treated Tiger Woods, Olympic swimmer Dara Torres and several other high-profile athletes.
The report of Rodriguez's ongoing investigation comes days after the 17-year vet was asked about the presence of his cousin Yuri Sucart, who was also involved in obtaining steroids for Rodriguez years ago.
Sucart was spotted in the Yankees' team hotel in San Francisco despite being banned from all team-related locations.
"There's no rules that are being broken," Rodriguez said Friday. "The team is fully aware. He stays at his own place, he's not on the team bus, he's not on team planes, he's not in the clubhouse, he's not in the stadiums, so we're complying with all the rules. One thing I learned a few years ago is my responsibilities, and they're very clear."
ESPNNewYork.com's Wallace Matthews contributed to this report.