Barr wouldn't elaborate on who exactly had contacted Carmona, but added that thus far he hadn't been in touch with national Democrats about a potential run. He added that Arizona Democrats believe that Carmona at the top of the ticket could boost other statewide Democratic candidates in 2014, when Arizona will see a state attorney general, secretary of state and state treasurer race, as well as the gubernatorial race, play out.

Carmona posted a surprisingly strong showing for a Democrat running in a red state against then-Rep. Jeff FlakeJeffrey (Jeff) Lane FlakeGOP strategist donates to Alabama Democrat Sasse: RNC help for Roy Moore 'doesn't make any sense' Sasse calls RNC decision to resume support for Moore 'bad' and 'sad' MORE (R), who ultimately toppled him. But Flake only won by 3 percentage points, while the Republican presidential nominee won the state by 9 percentage points, indicating Carmona had high bipartisan appeal.

He's still months away from a decision, but Barr said Carmona's staff is working to remain prepared in case the Democrat does decide to launch a bid.

"His staff is doing everything we can to keep the infrastructure ready in case he chooses to run," he said.