The idea of Mongolia joining the USSR does seem to have been floated from time to time--more by Mongolians than by the Soviets:
"Given Mongolia's profound dependence on the Soviet Union, Mongolians had several times proposed that Mongolia join the Soviet Union, yet Soviet leaders, wary of accusations from China, were not supportive. In the late 1920s, radical western Mongols...resented Khalkha domination and proposed that western Mongolia and Tuva together join the Soviet Union. In the 1940s and early 1950s the Soviet-trained technocrats under Choibalsang repeatedly questioned whether socialism could be built in Mongolia without joining the Soviet Union. The procurator B. Jambaldorj raised the possibility in 1944, when Tuva joined the Soviet Union, and Daramyn Tomorochir and Yumjaagin Tsedenbal raised it again late in Choibalsang's life. Choibalsang himself violently opposed such ideas, but after his death the Mongolian Politburo in 1953 approved unification, only to be rebuked by V. M. Molotov for their 'simple-minded error.' In the mid-1970s the Soviet ruler Leonid Brezhnev sounded out his Mongolian counterpart Tsedenbal about this issue. By then, however, the very success of Mongolian industrialization with Soviet aid had decreased Mongolia's perceived need for unification, and the issue was dropped." Article "Soviet Union and Mongolia," p. 515 in Christopher P. Atwood,
Encyclopedia of Mongolia and the Mongol Empire (New York: Facts on File, 2004).
http://shora.tabriz.ir/Uploads/83/cms/user/File/657/E_Book/History/Encyclopedia of Mongolia and Mongol Empire.pdf
As for why the Soviets were generally cool to the idea, I think the words "wary of accusations from China" sum it up.