We consider a generalization of Piatetski-Shapiro sequences in the sense of Beatty sequences, which is of the form ${\left(\lfloor \alpha n^c+\beta \rfloor \right)}_{n=1}^{\mathrm{\infty }}$ with real numbers $\alpha ⩾1,c>1$ and $\beta$.In this paper, we prove that there are infinitely many $R$-almost primes in sequences ${\left(\lfloor \alpha n^c+\beta \rfloor \right)}_{n=1}^{\mathrm{\infty }}$ if $c\in (1,c_R)$ and $c_R$ is an explicit constant depending on $R$.