St. Jerome mentions the correspondence in his De Viris Illustribus, XII:
Lucius Annaeus Seneca Cordubensis, Sotionis Stoici discipulus, et patruus Lucani poetae, continentissimae vitae fuit, quem non ponerem in catalogo Sanctorum, nisi me illae Epistolae provocarent, quae leguntur a plurimis, Pauli ad Senecam, et Senecae ad Paulum. In quibus cum esset Neronis magister, et illius temporis potentissimus, optare se dicit, ejus esse loci apud suos, cujus sit Paulus apud Christianos. Hic ante biennium quam Petrus et Paulus coronarentur martyrio, a Nerone interfectus est.
The above Wikipedia article cites Lightfoot, Dissertations on the Apostolic Age, who categorically denies their authenticity:
The poverty of thought and style, the errors in chronology and history, and the whole conception of the relative positions of the Stoic philosopher and the Christian Apostle, betray clearly the hand of a forger.
It may be a tall order, but an ideal answer could list some concrete points that establish Lightfoot's thesis. To make it even more clear, I'll bullet-list the three general categories mentioned above as questions:
- What style choices make it clear that Paul/Seneca did not write the respective epistles?
- What chronological/historical errors are there?
- What ideas are present that do not make sense for a Stoic philosopher or an early apostle to voice?
Partial answers (as long as they have concrete points) are encouraged!