June is not a preferable time (climate wise). I would also recommend you to keep a check on weather details before visiting.
if there is raining possibility at the time of your visit, weather can turn out to be pleasant
Climate wise Bihar is pretty hot during that time. If it rains, the weather becomes pleasant. Therefore, you can visit there first week of July. Both Climate and Greneries of the hill top areas will be a treat.
If you are planning to go to Bodhgaya, you can visit Nalanda and Pawapuri as well. Beginning of monsoon is really perfect time to visit Bihar.
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