FDR was understandably frustrated by the refusal of the SCOTUS, led by the "Four Horseman" -- four conservative justices -- to affirm his New Deal agenda. So in 1937 he came up with a plan to reshape the court by appointing a new justice for every justice over the age 70. This would have given him six new appointees. However, the only way to implement this plan was to get Congress to enact it in legsislation. FDR proposed such legislation and it blew up in his face. The Senate Judiciary Committee refused to report the bill favorably and the Senate as a whole, despite a very lopsided Democratic majority (76-16) voted to kill the bill by a overwhelming 70-20 vote.
While it is sometimes argued that FDR ultimately succeeded because his court packing plan caused the court to tilt in his favor, what really made the difference was that three of the conservative Four Horseman retired or died within two years of the unveiling of the court packing plan and FDR was able to replace them with justices more likely to support his agenda. Indeed, the first retirement, that of Justice Willis Van Devanter, occurred even before the Senate voted on FDR's plan.
Given that history, a proposal by President Obama for legislation altering the Court's make up would blow up in his face. There is zero chance that such legislation would be approved by the Senate and it would merely link Obama with what is generally understood to be one of FDR's biggest failures.