The text employs the word "jealous" in relation to Louis and Philip but not to Eleanor. If Eleanor is to be seen as jealous, it has to be inferred indirectly from the text. Jealousy is defined as resentment over the success of a rival. For Eleanor to feel jealous of Thomas Becket, she would have to view him as a rival for power. Though she balks at Becket arranging the engagement between Henry and Marguerite, she says it is because she wanted "to arrange a great pageant to impress" Louis and his court. She also concedes that King Henry is correct in authorizing Becket for the task because of the bitter feelings Louis and "the Franks" have toward Eleanor.
While Eleanor is insistent upon preserving her power, authority, and influence--and insistent on having them duly noted--she stands up to Becket in a queenly fashion that belies the suggestion of resentment that would mark him in her mind as a rival: "And that, Henry, is as much due to me as it is due to Becket." [On the other hand, there is strong evidence that Louis threatens Eleanor: "despite all her bravado, Eleanor's feelings were hurt" by Louis.]
While Eleanor may not feel jealousy toward Becket as a rival, she does not like him. As she tells Mother Matilda, "I would like him more if Henry liked him less." This may be interpreted as her dislike of his influence upon Henry rather than jealousy toward Becket. It may seem there is small difference, but in jealousy the emphasis is on self, whereas concern over influence is directed away from self to others: it relates to external exertion of power to influence what appears to be wrong directions or wrong decisions. This interpretation is supported by the conversations between Matilda-Empress and Eleanor when they are "Up":
(Eleanor) People blamed Henry for Becket's murder. Young Henry blamed him more than anyone else; after all, Thomas had been as much a father to him Henry had been.
Eleanor's conversation here, like others, has no trace of self-focused jealousy. Therefore, according to the text, Eleanor was not jealous of Becket though she did concern herself over the influence Becket did or might have over Henry: she did not like Becket but was not jealous.