Joe Biden's advisers think that Sen. Joe Manchin (D-WV) ultimately will decline to run for president on the No Labels ticket. They base this on his love of publicity and staying in the limelight as long as he can, but in the end coming home and supporting Biden. Manchin also knows that if he ran on the third-party ticket, that would probably help elect Donald Trump. While Manchin gets along with some Republicans, they are typically Sens. Mitt Romney (R-UT), Susan Collins (R-ME), and Lisa Murkowski (R-AK). He doesn't especially get along with and make deals with the Trumpiest Republicans.
Despite Manchin frustrating Biden at times, he has voted for nearly all of Biden's nominations at all levels. He also has a history with Biden. During the Obama administration, then-Vice President Biden was the only top-level official who talked to Manchin. They bonded over their working-class upbringing and their support for unions (in Manchin's case, primarily the coal miners' union). Although Manchin didn't give Biden a heads up about his decision not to run for reelection, Biden didn't expect him to run because he would be running against a popular sitting governor in a hugely pro-Trump state.
Manchin has remained in contact with the No Labels leadership, keeping his options open. One of the issues is that Manchin is concerned the group doesn't have enough resources to run a modern presidential campaign. He doesn't want to run a shoestring campaign whose only function is to defeat Biden.
Another indication that Manchin is not serious about running for president is that he has said he will make a decision after Super Tuesday. By then he would probably know who the Democratic and Republican nominees were. But organizing a presidential campaign starting in March would be extremely late in the cycle. If he were serious, he would want to start months earlier. In fact, by dragging things out, he makes it tough for No Labels to run anyone with any efficacy, since they are going to wait on Manchin, and then, if and when he jumps ship, they'll be left scrambling. Maybe, just maybe, this semi-dalliance is one last gift from Joe M. to Joe B. (V)