So, naturally, when the Republican former Florida governor and former secretary of state took the stage at a higher education conference in Dallas today, albeit a few hours apart, the level of speculation shifted into overdrive.
But they never appeared on stage together, shook hands for the cameras to see or placated the crowd with lighthearted joking about the potential for a 2016 match-up.
Instead, the two possible future presidential candidates focused on the challenge of expanding access to higher education at home and abroad.
Clinton was received by rousing applause when she took the stage after a highlight reel of her advocacy on education in Arkansas, as first lady, and as secretary of state, and after a glowing introduction by co-host and former North Carolina Gov. Jim Hunt.
And only briefly did she tantalize the crowd by thanking Bush, the event's co-host, for his dedication to education during his time as Florida governor.
"I want to thank Jeb Bush, someone else who really focused on education during his time as governor," Clinton said. "And who has continued hat work with passion and dedication in the years since."
And with that, the moment passed.
Her speech was dedicated to an issue Clinton has made a signature feature of her public life: expanding educational opportunity, especially for women and girls.
Clinton worried that access to higher education is eroding at home, a concern Bush shared in his brief introductory remarks.
"I worry that we're closing the doors to higher education in our own country," Clinton noted. "Providing education for everybody has to be the goal, and not just any education but a quality education."
Earlier, Bush noted that exporting U.S. higher education can help resolve dual problems of America's "growing affordability problem" and the world's struggle with "accessibility."
And Clinton referenced Pakistani schoolgirl Malala Yousafzai, who was shot by the Taliban for advocating that girls be permitted to get an education.
"When they couldn't shut up Malala because she had learned to speak for herself, they tried to kill her," Clinton said. "Her story reminds us that far too many women and girls are denied opportunities to develop their own God-given potential."
This isn't the first time Bush and Clinton have crossed paths in recent months.
Bush presented Clinton with the National Constitution Center's Liberty Medal in Philadelphia last year and he addressed the awkward speculation head on.
"Hillary Clinton and I come from different political parties, and we disagree about a few things," he said at the September event. "But we do agree on the wisdom of the American people, especially those in Iowa, and New Hampshire and South Carolina."
They have both said their decision to run for president in 2016 or take a pass will come by the end of this year.
But if they do choose to run, both could be trailed by doubts about whether the country really wants to have another dose of "Clinton" or "Bush" leadership for four more years.
If anything, however, the ability of Bush and Clinton to share the stage in Dallas is a unique feature of their mutual nonofficial lives.
They can both dedicate dozens of speeches and appearances to their pet causes, largely sidestepping the political landmines of the day.