Senate Minority Leader Mitch McConnell (R-Ky.) is saying thanks but no thanks to President Barack Obama’s last-minute offer to bring another Republican Senator to Thursday’s bipartisan health care summit.

The president’s offer comes after the White House on Wednesday sidestepped McConnell and directly invited moderate Sen. Olympia Snowe (Maine), a key Republican health care negotiator through much of last year. Snowe declined the invitation.

But the move irked Republican leaders, who argued the White House was breaking its own detailed rules for the nationally televised event — as outlined in the invitation sent to Congressional leaders. Snowe supported the Finance Committee health care reform bill and was the only Senate Republican to vote in favor of a Senate health care bill last year.

“They made the extra invites after claiming that there was no room available for additional Members or staff,” a senior Republican Senate aide said Wednesday evening.

The White House also extended an extra, direct invite to Sen. Ron Wyden (D-Ore.), who has focused on health care policy throughout his Congressional tenure. Wyden accepted.

Obama’s summit invitees include top Congressional leaders and the relevant committee chairmen and ranking members. The White House also allowed the four leaders to invite four Members each of their own choosing.

McConnell’s four invitees are GOP Conference Chairman Lamar Alexander (Tenn.); Sen. John McCain (Ariz.), the 2008 Republican presidential nominee; and Sens. John Barrasso (Wyo.) and Tom Coburn (Okla.), both doctors.