For a prime p and a positive integer n, an integral quadratic form over a ring ℤ_p is called primitively n-universal if it primitively represents all integral quadratic forms of rank n over ℤ_p. In 2021, Earnest and Gunawardana provided criteria to determine whether a given integral quadratic form over ℤ_p is primitively 1-universal. In this talk, we prove that the minimal rank of primitively n-universal integral quadratic form over ℤ_p is 2n, if p is odd or if n is at least five. Moreover, we obtain a complete classification of primitively 2-universal integral quadratic forms over ℤ_p of minimal rank.