President Barack Obama’s claim as the first “Web 2.0 president,” along with his ties to the tech industry, may be getting a little deeper with rumors of two impending personnel moves between his presidency and Silicon Valley.

Obama, of course, was not the first politician to use Web 2.0 technologies to reach voters, but he, and his team, can be credited as the first to fully capture the viral powers of Facebook, Twitter and YouTube to reach voters and raise money.

Once elected, he looked to the CEOs and founders of those sites, and others like Google, for advice – hosting them at dinners, recruiting them to serve on advisory boards and including them in policy discussions.

Now it appears that partnership is moving from the counsel room to the HR department.

The New York Times is reporting that social networking giant Facebook is courting former White House spokesman Robert Gibbs to join its policy and communications team in a “senior position.”

If Gibbs were to accept a job with Facebook he could cross paths in mid-air en route from Washington, D.C. to Palo Alto, Calif. with former Google CEO Eric Schmidt as he heads the opposite direction, from Googleplex to the White House.

Schmidt is reportedly on the short list to become the next U.S. Secretary of Commerce. Schmidt would replace Gary Locke who has been nominated by the president to be U.S. ambassador to China.

Schmidt announced in January that he would leave Google after more than a decade, and his last day as CEO of the search giant is fast approaching.

Schmidt will step down this Monday, April 4th and has not announced any future plans.

Discussions between Facebook and Gibbs, who left the White House in February to, it was expected, go on the paid speaking circuit and work on the president’s reelection campaign, are in their early stages and could still fall apart, according to The Times.

But if Gibbs were to join Facebook, he would see at least one familiar face in the office cafeteria.

Last June, Facebook recruited and hired the former chief of staff of the White House National Economic Council, Marne Levine, as its vice president for global public policy.

The path between Washington, D.C. and Silicon Valley, and vice versa, is not all smooth sailing, however.

Privacy watchdogs have heavily criticized and fought back against rumors of Schmidt’s possible Commerce Department nomination.

Consumer Watchdog, a well-known thorn in Schmidt’s side, issued a press release objecting to a possible appointment, and sent a letter to Obama asking him not to nominate Schmidt for the position.

“Putting Eric Schmidt in charge of policing online privacy is like appointing Bernie Madoff to direct the Securities Exchange Commission,” the release said.

Meanwhile, followers of both politics and Facebook are already questioning the potential Gibbs’ hire.

“Who better to manage communications for a global company that rarely answers a direct question about its management than the former spokesman for a White House that rarely answered a direct question about its management? One wishes Facebook wasn’t so interested in such a conventional choice…,” commented Micah Sifry on his techPresident blog.

And Politico was quick to question if Gibbs is even “cool enough” to work at Facebook, pointing to this exchange with reporters:

“In a press briefing in February 2010, a reporter asked Gibbs about White House officials “grumbling” because they had to give up using social media, like Facebook and Twitter, for security reasons. 

”Bill was probably on Facebook,” Gibbs said, referring to the deputy press secretary Bill Burton. “I’m not nearly as young and hip as [him].” “Clearly,” the reporter laughed.”

What do you think? Is the path from Silicon Valley to the White House, and vice versa, a useful one for social media?