No, USSR did not have proper aircrafts to perform such reconaissance. The closest thing to U-2 was an M-17 (and its later version M-55, which were already useless as spy gear by the time they were issued. It seems that Soviet Union understood that by mid 60's the future was after spy satellites and didn't want to waste resources on designing a potentially obsolete vehicle. They also had very limited opportunities with routes where they could fly, becase there weren't many USSR-friendly countries around USA (except Cuba, whose strategic position is questionable). USA, on the other hand, had plenty of route planning opportunities, e.g. the notorious Powell's U-2 was scheduled top fly from Peshwar in Pakistan to Bodo in Norway.

The only aircrafts comparable in role to SR-71 were MiG-25R/RM/RBN/RBT/RBV. But since MiG-25 was designed as an interceptor, it had fairly limited range, i.e. couldn't be operated far from bases. As spy planes they were used during the Israeli occupation of Sinai Peninsula between Six-Day War and Camp-David Accords and during Iran-Iraq War. In both cases their main advantage was such high altitude that they were out of range of any AA or SA missiles, a couple of Soviet pilots were even decorated for this trick.