The correlation between mechanical stress and magnetic anisotropy of Ni and Fe films on Cu(001) is investigated. The magnetoelastic coupling and the film stress during the growth are measured in situ with a highly sensitive optical bending beam technique. For Ni a dramatically reduced magnetoelastic coupling of B-1=3.5 MJ/m(3) is found for films thinner than 10 ML, roughly one third of the bulk value of 9.4 MJ/m(3). This change is explained by a strain correction to the magnetoelastic coupling. The influence of the interfaces does not significantly contribute to the magnetoelastic coupling. A very small magnetoelastic coupling of 0.4 MJ/m3 for Fe films in the range from 12 ML to 25 nm is attributed mainly to the crystallographic orientation of the bcc-Fe.