Three explosions rocked the eastern provincial capital Jalalabad on Saturday in attacks targeting Taliban vehicles.
There was no immediate claim of responsibility, but Islamic State militants, headquartered in the area, are enemies of the Taliban, the report said.
Islamic State offshoot ISIS-K claimed last months' bomb attack on Kabul airport that killed more than 170 persons, including 13 US Marines.
Three injured in the blast were civilians and 16 were Taliban fighters, some of whom are in a critical condition.
Also on Saturday, a sticky bomb exploded in Kabul, wounding two. The target of the bomb was not immediately clear.
The Taliban are facing major economic and security problems as they attempt to govern, and a growing challenge by IS insurgents would further stretch their resources, the report said.