The notion and some properties of (strongly) B-rings, in a natural way, are extended to (strongly) B- and (strongly) BJ-semirings which is somewhat similar to the notion of rings having stable range 2. Results are given showing the connection between several types of semirings whose finite sequences satisfy some stability condition, some involving the Jacobson k-radical of the semiring R. Besides some examples and other results, it is shown that R[x], the semiring of polynomials over a semiring R, is not a B-semiring (consequently, not a strongly B-semiring) when R is a zerosumfree semiring. We also study some algebraic properties of the S-relative B- and BJ-semirings with respect to a nonempty subset S of R.