Donald Alfred Hagner, The Use of the Old and New Testaments in Clement of Rome. (Series: Supplements to Novum Testamentum, 34). Leiden: E.J. Brill, 1973.

The possible allusions to 1 Timothy in Clement's epistle are numerous. Individually they are not very convincing; taken together, however, they establish a probability that Clement knew and was influenced by 1 Timothy. (Hagner, 232)

This is followed later by:

... A common ethical catechesis may well account for a number of the parallels. However, even when allowance is made for such agreement, it seems improbable that all the allusions are to be explained in this way. Thus on the basis of the evidence cited, our conclusion is that Clement probably knew and made use of 1 Timothy and Titus; for Clement's knowledge of 2 Timothy, however, the evidence is less convincing and justifies not more than a conclusion of possible dependence. (Hagner, 236)

I deal with a limited set of possible allusions to the Pastorals in First Clement over on PastoralEpistles.com; though I should obviously sift through Hagner's work (and Lightfoot, of course, though I'm guessing Hagner has already mined that work) and supplement that list.