The author generalizes a result of \textit{G. Huisken} and \textit{C. Sinestrari} [`Mean curvature flow singularities for mean convex surfaces', Calc. Var. Partial Differ. Equ. 8, No.1, 1-14 (1999)], giving conditions such that under the mean curvature flow a type II singularity develops. Whereas Huisken-Sinestrari's paper gets the result assuming that the compact initial hypersurface is mean convex, the present paper poses a priori estimates on a whole family of closed hypersurfaces evolving by mean curvature. The proof uses essentially the same methods as Huisken-Sinestrari's paper. The second part of the paper shows that the a priori estimates are satisfied for mean convex hypersurfaces and some variety hereof and for starshaped hypersurfaces.