Simpson said he recently became aware that Ask.com provides a service called "AskEraser," which allows site users to their personal information from the company's systems.

"That's what prompted us to say [to Google], if these guys can do it, why can't you?" Simpson said.

Consumer Watchdog is also asking Google chairman and CEO Eric Schmidt for an in-person meeting to discuss their request.

"I'm very optimistic that he's sincere about listening to people's concerns," Simpson said. "I fully expect we'll have a meeting at some point."

Google did not immediately respond to a request for comment Friday.