The plan to resume production of Tu-160 bombers is alive and well. Reporting on a visit of the deputy defense minister to the Kuznetsov plant in Samara, the ministry of defense press service said that preparations are underway, although not everything is going smoothly. The plan is to begin production of new aircraft no later than 2023. The new bomber, while still Tu-160, is expected to be almost a new plane, which is not surprising given that the design is more than 30 years old. There was no word on the number of planes that will be produced this time.
Meanwhile, Russia continues to upgrade its current Tu-160 fleet. In December 2015, the Gorbunov Aviation Plant in Kazan completed an overhaul of the Vasily Senko bomber. According to the CAST blog, it's the fifth Tu-160 aircraft that underwent deep modernization:
2006: Valentin Bliznyuk, 19 red, RF-94113, serial number 2-02
2011: Valery Chkalov, 17 red, RF-94110, 5-04
2014: Andrey Tupolev, 18 red, RF-94111, 6-05
2015: Vasily Reshetnikov, 02 red, RF-94102, 7-02
2015: Vasily Senko, 11 red, RF-94114, 6-02
This appears to be the process of deep modernization that is supposed to be completed by 2019. Or maybe not - it appears that some bombers go through a different kind of overhaul. For example, neither Aleksei Plokhov, which returned to service in 2009, nor Ivan Yarygin, returned in 2010, is not on the CAST list.
UPDATE: As my readers point out, the CAST list is not necessarily correct. And there are indeed, several Tu-160 modernization "streams" going on. Here is a good (if somewhat old) take on the complicated situation.