If "strict fibre" just means the fibre over *, then you can make the construction as follows. There are two steps.
First, let f:X->X be homotopic to the identity and contract some
neighborhood N of *
onto *. Pulling back your fibration p through f, you are reduced to the case where
p is a projection FxN->N over N. Then, consider a function g: N->I whose value is 0 on partial N and 1 on *; [consider the mapping cylinder C of a homotopy
equivalence F->F' and the projection pr:C->[0,1] with pr^-1(1)=F']; define E' over N
as the amalgamated product of C with N over G; define E' as E over (X-N). if you have chosen N to retract by deformation on *, then this E' will have the property you want.
Maybe there is a more elegant way to present the construction (i'm not an expert in homotopy theory nor cw-complexes)
Sorry, I come back to this answer which I realize is not quite correct, because C is not a Serre fibration over I=[0,1], in general.  Instead of C, you should use a Serre fibration over I whose fibres over 0 and 1 are F and F' respectively; there should be a canonical construction of this kind, but right now I cannot be more precise. Hope it helps.