House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) told President Reagan yesterday that he hoped Reagan's proposed bipartisan study commission on Social Security could report back to Congress by April 15 on steps needed to assure the system's solvency.

O'Neill said Senate Majority Leader Howard H. Baker Jr. (R-Tenn.) agreed on that target date, which would produce a report well in advance of the 1982 elections.

Reagan had mentioned the possibility of a one- or two-year study. But O'Neill, in a letter released yesterday, said no "additional extensive analysis" was needed.

O'Neill and House Majority Leader James C. Wright (D-Tex.) have repeatedly made clear they view the president's proposal for a bipartisan commission as an attempt to give a nonpolitical coloration to a group that in fact would be White House-dominated with five presidential appointees, five appointed by Baker and five by O'Neill. They think the president is trying to make the Democrats share the heat involved in cutting Social Security.

In an attempt to reduce charges of White House domination, it has now been agreed that O'Neill and Baker each will make three appointments, one from Congress and two from outside; and that Senate Minority Leader Robert C. Byrd (D-W.Va.) and House Minority Leader Robert H. Michel (R-Ill.) each will make two appointments, one from Congress and one from outside.