Yes, almost immediately (a month), according to Letters & Papers. But Henry wouldn't nibble this time. Catherine Howard was executed in February 1542, and in March 1542 Marillac reports "He seems very old and grey since the mishap (malheur) of this last Queen, and will not yet hear of taking another, although he is ordinarily in company of ladies, and his ministers beg and urge him to marry again." [emphasis mine]The Emperor writes to Chapuys the same month to know "if Henry is inclined to marry again, and in what quarter. Chapuys must try and prevent his taking back Anne of Cleves." There seems to have been a general expectation, both at home and abroad, that Henry would get married again. The Emperor was against Anne of Cleves being reinstated, as her brother was allied with the French; and Chapuys responds dutifully in April that he "will do his best to prevent the King taking Anne of Cleves again; but as yet there is no appearance of it. Indeed, except that he frequents ladies' company for mirth, as a man nurtured among them, he seems not to think of a new marriage. He has been low-spirited ever since he heard of the late Queen's misconduct."The French apparently felt that if Henry was not interested in Anne of Cleves, there were plenty of other French-backed fish in the sea. In June Chapuys reported that Henry had said "the French were continually offering him ladies to marry," perhaps some of the ones they originally offered after Jane Seymour died, although no names are specified. "Chapuys pleased him by answering that no doubt they would do as they did in the case of the Princess, now Queen of Scotland [Mary of Guise], for in marriages they always broke treaty with England ..."Most of the diplomatic correspondence at the time centers on the prospective marriage of Mary Tudor to the Duke of Orleans. The negotiators had a fondness for matrimonial package deals - Henry and his daughters and son being matched with suitable spouses on the other side - so perhaps the French were hoping to secure Henry as well as Mary.I couldn't find the Imperials making an offer to Henry, merely the focus on blocking Anne of Cleves. Part of the difficulty is that the Emperor's female relatives were also related to Henry through the affinity created by his relationship with Catherine of Aragon. They would need a Papal dispensation to marry him, and Henry repudiated Papal authority, claiming a dispensation issued by Cranmer was enough.
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