In reality Henry had no grounds for an annulment, since the dispensation granted by Pope Julius to Henry VII and the Catholic monarchs covered all bases. Best shot would have been forcing/coaxing Catherine into a vow of celibacy.
That said, Henry was a very loyal son of the Church and would have got his way had Rome not been sacked. Clement's suggestion was that he marry again and have Wolsey as legate declare his marriage to Catherine invalid; when the matter was deferred to Rome Clement would give a favourable ruling and any children would be legitimate since the new marriage was contracted in good faith.