Ross met with Tomblin in the Capitol last week. He also gave $5,000 to Tomblin's inauguration committee.
Hawse said he is a Democrat with a proven record of winning elections.
"I've never lost, and on more than one occasion I have led the ticket," Hawse said.
But one name that is unlikely to be on the list is making noise, too.
Randolph County Democrat Margaret Kerr Beckwith said she, above all others, deserves the seat. Last year Beckwith challenged Barnes, who soundly defeated her 24,571 votes to 12,941.
"I ran and in the primary I was one of only three women from the Democratic Party that ran for the West Virginia state Senate," she said.
Beckwith said she is not getting consideration because she lives in Randolph County, where Barnes is from.
But she said Barnes is actually spending much of his time in Upshur County.
Barnes just sold nearly eight acres of property in the Randolph County town of Beverly, according to the Randolph County deputy clerk.
Barnes said there are three homes on the farm and he still lives in one of them.
Beckwith also questioned Ross' decision to change his registration to Upshur County.
Ross said his office and business are there.
"We can't use offices and old houses we are going to fix up someday," Beckwith said.
From her perspective, Barnes actually lives in Upshur and Ross actually lives in Randolph, but Barnes is pretending to live in Randolph and Ross is pretending to live in Upshur.
Barnes denies this.
"Obviously she hasn't out been out to my place in Beverly for coffee in a while, but she's welcome to come out; just give me a call," he said.
Barnes said he does spend some time overnight in different locations, including Lewisburg, Buckhannon, Fairmont and Kingwood.
The search committee is unlikely to consider Beckwith without legal proof of her allegations.