After the death of John Carmel Heenan (who was a close friend of Dwyer's) in 1975, Dwyer was seen by some as a natural successor as Archbishop of Westminster. However, Dwyer informed the Apostolic Delegate he felt that at sixty-seven his age was too great for him to be considered for the post. He was, however, elected president of the Bishops' Conference during the first three years of Basil Hume's episcopate, becoming the first bishop to hold that position who was not also Archbishop of Westminster.