Blackstone Group LP (BX) and activist investor Carl Icahn submitted proposals to buy Dell Inc. (DELL) that would rival a $24.4 billion buyout offer from Silver Lake Management LLC and company founder Michael Dell, according to people with knowledge of the matter. Blackstone, the world’s biggest private-equity firm, outlined an offer valued in excess of $14.25 a share in cash, said one of the people, who asked not to be identified because the process is private. Blackstone is working with Morgan Stanley, which is confident it can raise debt to finance the deal, according to the person. Icahn said he’d pay $15 a share for a leverage recapitalization with a 58 percent cap on the amount of cash used in the deal, the person said.