Yes this actually happened. Eisenhower was one of several individuals who talked Churchill out of joining the D-Day force, although it was the King who actually convinced Churchill not to go.. By late May 1944 Churchill had decided that he wanted to be on one of the warships - HMS Belfast - while it bombarded the Normandy coast on D-Day. While the King initially had no problem with this and in fact wanted to accompany Churchill.

The King's Private Secretary, Sir Alan Lascelles, the Naval Commanded in Chief of the Allied Expeditionary Force, Admiral Bertram Ramsay, and Eisenhower were strongly opposed to either of them going.^(1)

Their objections convinced the King that neither he nor Churchill should attend although Churchill still insisted on going even if the King refused. On the 2nd of June 1944 the King wrote to Churchill:

I want to make one more appeal to you not to go to sea on D day. Please consider my own position. I am a younger man than you, I am a sailor, & as king I am the head of all three services. There is nothing I would like better than to go to sea but I have agreed to stay at home; is it fair that you should then do exactly what I should have liked to do myself? You said yesterday afternoon that it would be a fine thing for the King to lead his troops into battle, as in old days; if the King cannot do this, it does not seem to me right that his Prime Minister should take his place. Then there is your own position. You will see very little, you will seem a considerable risk, you will be inaccessible at a critical time when vital decisions might have to be taken, & however unobtrusive you may be, your very presence on board is bound to be a very heavy additional responsibility to the Admiral & Captain...I ask you most earnestly to consider the whole question again, & not let your personal wishes which I very well understand lead you to depart from your own high standard of duty to the State.^(2)

Churchill did not immediately reply to this letter as he was already on his way to Portsmouth. Lascelles called his train and managed to get Churchill to verbally agree that he would not go. He was not very happy about it though and on 3rd June 1944 he wrote back to the King:

as Prime Minister and Minister of Defence, I ought to be allowed to go where I consider it necessary to the discharge of my duty... I rely on my own judgement, invoked in many serious matters, as to what are the proper limits of risk which a person who discharges my duties is entitled to run. I must most earnestly ask Your Majesty that no principle shall be laid down which inhibits my freedom of movement when I judge it necessary to acquaint myself with conditions in the various theatres of war. Since Your Majesty does me the honour to be so much concerned about my personal safety on this occasion, I must defer to Your Majesty’s wishes and indeed commands.^(3)

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Sources

^(1) Andrew Roberts, Churchill: Walking with Destiny (2nd Edition: Penguin, 2019), p.819

^(2) Ibid., pp.819-820

^(3) Ibid., p.820