Is it possible to conduct exploratory factor analysis using 'fixed number of factors' instead of going for 'factors having Eigenvalues more than 1'? Are there any technical problems in that?
(This is possibly a comment rather than an answer but is long)
To assume a fixed number of factors a-priori usually means to be on the side of testing/confirming hypotheses, for instance by a chi-square test, and so, by common use of language means to be not "on the exploratory side".
If we don't want to assume simply an error in the use of statistical terms - what could then be the "exploratory" part after the number of factors has been assumed (and we have a framework to which we can approximate by a least-square-error model)? The type of rotation? The orthogonality-assumption at all?
It sounds like you a hypothesis about the factor structure a priori, so why not test it within a CFA? If your intention is to understand other aspects of the analysis that might be revealed in modification indices then maybe do an EFA in a CFA framework.