It is very likely that Harold Godwinson pledged some sort of oath to William the Conqueror in the early 1060s. Although, the scope of the oath is up for debate. Harold probably swore some sort of oath of fealty to William and promised to help him win the throne of England when the Edward the Confessor passed, but Harold -- as an earl -- could not outright promise the throne of England to William; that was the privilege of the king and the Witenagemot (a council of nobles who elected the English monarchs). Harold's support would all but assure William's ascension to the English throne, but he couldn't explicitly pledge him the throne in the exact way that the question implies.

The key to this question, though, is duress in regards to Harold's oath. As I'm sure you know from your comment below, Harold Godwinson was shipwrecked on the Continent in either 1064/65 (we're still not 100% sure of the exact year) and was captured by a certain baron named Guy. Guy then transferred Harold to his overlord, William the Bastard, Duke of Normandy.

From this point forward, Harold was William's captive and at his mercy. The Bayeux Tapestry depicts the two magnates as good friends, and they may indeed have behaved outwardly friendly towards each other, but the reality was that Harold knew his life was endangered by William. Given this reality,Harold was shrewd enough to know that he only had one option: do anything William asked of him, including swear an oath. He could always attempt to annul it anyway, it's not like this was a decision taken freely. The fact that the Bayeux Tapestry plays up this oath, publicly sworn on the bones of saints, and that the Conqueror used this oath to portray Harold as a perjurer in his appeal to the Papacy and the collective conscience of Europe, underlies its historicity. If Harold had sworn no oaths, he would have vigorously denied it and William would not have relied so heavily on it for propaganda purposes. The English, in turn, pointed to the fact that the oath was made under duress, as Harold was William's prisoner at the time.

Sources:

Bayeux Tapestry

William the Conqueror by D.C. Douglas

The Norman Conquest Revisited by Richard Huscroft

Just to clarify here, are you sure you don't mean Edward the Confessor? Because as far as I know, he's the one who was meant to have promised the English throne to William.