Bek and his elder brother Thomas Bek were members of a family of knights. Their father was Walter Bek, who held lands at Eresby in Lincolnshire. Another brother was John Beke, 1st Baron Beke of Eresby, who held the family lands in Eresby. Walter had married Eve, the niece of Walter de Gray, Archbishop of York. Bek was born about 1245 and was Walter’s third son. Both Thomas and Antony were educated at Oxford University, where they studied from 1267 to 1270.
Having entered the clergy, Bek received several benefices and soon attracted the attention of the Lord Edward, the heir of King Henry III of England. He was Archdeacon of Durham by 1275, as well as precentor of York and held prebends at Lichfield, London, and Wells. Bek accompanied Edward when the prince went on Crusade in 1270. When Edward returned from crusading, he appointed Bek to the office of Keeper of the wardrobe on 21 September 1274, but Bek held the office for only about a month, and was replaced in October with his brother Thomas Bek. Edward named Bek the constable of the Tower of London in 1275. In November 1277 Bek was one of the councillors sent by Edward to negotiate with Llywelyn ap Gruffudd, Prince of Wales. The result was the Treaty of Aberconwy which required Llywelyn to give up hostages, pay a large fine, give up claims to some lands and pay homage to King Edward. Bek was also sent to Aragon to negotiate the marriage of Edward’s daughter Eleanor to the heir of Aragon, Alfonso.