By Lauren Fox, Ted Barrett and Elizabeth Landers

CNN

WASHINGTON (CNN) -- Senate Republican leaders continue to face criticism for appointing 13 men to a working group on health care and not including a single woman in the negotiations.

Democrats continued to grill the GOP on their handling of health care Tuesday.

Sen. Patty Murray of Washington told reporters, "We are going to keep pushing Senate Republicans to take a different path than the House did and sending a very clear message to the 13 Republican men currently negotiating a secret health care plan, which I really hope is not happening in the men's locker room at the very least."

The official working group, CNN reported Friday, failed to include any women, despite the fact several female GOP senators have been involved in health care legislation or could be key votes to getting the bill passed.

Sens. Susan Collins of Maine and Lisa Murkowski of Alaska are two such senators.

Collins said she will continue to work on the issue regardless.

"It doesn't bother me. The leaders have a right to choose whomever they wish. It doesn't mean that I'm not going to work on health care," the Maine senator told reporters. "I've worked on health care for many years. I spent five years in state government overseeing the bureau of insurance many years ago, and I think I can bring some experience to the debate that will be helpful."

Collins is the co-author of legislation with Louisiana Republican Sen. Bill Cassidy, a former doctor who was also excluded from the working group. Collins told reporters that she planned to continue to work with Cassidy to build momentum for their health care proposal, which would repeal the individual and employer mandates, but would give states more flexibility to build their own health care systems.

There will be "a number of different groups" getting together on health care, Collins said, adding that she and Cassidy would continue to try and attract some "enlightened Democrats" to work with them.

Sen. John Thune, R-South Dakota, when asked why there is no woman in the group said, "well, I don't make that decision."

Asked if he would advocate for that he said, "I'm for it."

Thune spoke as he was headed into a meeting of the Senate working group in a room near Senate Majority Leader Mitch McConnell's office.

Sen. Shelly Moore Capito entered the meeting Tuesday morning, saying she had been invited to this session. But it doesn't appear she's been added permanently.

Senate Majority Whip John Cornyn, a Republican from Texas, also faced tough questions as to why someone like Collins -- who has been publicly involved in the health care debate -- wasn't included in the working group.

"She's an integral part of the discussion," Cornyn said, acknowledging she was not part of the official group last week.

Cornyn said the controversy was "totally bogus because every single woman in our conference is involved. We would invite Democratic women. We would invite everybody."

Asked about the lack of women on the working group, a senior White House official told CNN Tuesday there will be women added.

"You'll see these optics addressed," the official said, declining to name which female senators will join the group.

Senate leadership's current goal is to repeal and replace Obamacare using a process known as reconciliation, which would only require a 51-vote Senate majority to pass. The process is intended to bypass Democrats. Republicans hold a 52-48 advantage in the Senate.

"The GOP is crafting policy on an issue that directly impacts women without including a single woman in the process. It's wrong," California Democratic Sen. Kamala Harris tweeted Friday.

Texas GOP Sen Ted Cruz, a conservative who is on the working group, said it is "essential" Republicans include views from all comers. It's what "we've got to do to come together."

A GOP aide Friday defended the makeup of the group.

"We have no interest in playing the games of identity politics, that's not what this is about; it's about getting a job done," the aide said. "We'll work with any member of any background who wants to pass a health reform bill that will reduce premiums and take away the burdens that Obamacare inflicted."

CNN's Dana Bash, Ashley Killough and Elizabeth Landers contributed to this report.