I've looked at quite a few FBI and military files that were made on Manhattan Project members (I have a little collection of them), and the Jewish ones do get mentioned for being Jewish. Typically it seems to have been treated as a "compound" sort of thing; if they are suspicious for their political leanings, then it gets thrown into the reasons to be suspicious. Here's an excerpt from Philip Morrison's Army background investigation from April 1944:

Personal characteristics: Honest, discreet, uses sound judgment, reliable, dependable, aggressive, ambitious, trustworthy, capable, tactful, industrious, intelligent, brilliant, impressively bright, good talker, pleasant, polite, friendly, well-liked, studious, forceful, sincere, has good habits, neat, clean. (Memos A-1, A-O, A-P, A-Q, A-R, A-S, A-T, A-U, A-V, A-Y, A-Z, B, C, H, I, DD, F-1, F-2.)

Taciturn, outspoken, indiscreet, pushy, crude, has pronounced Jewish traits of asserting rights and demanding attention, bitter toward life, big talker, loud and boisterous, expounds a vast collection of facts but most of them incorrect when chased to their source. (Memos A-1, A-S, A-T, A-V, A-U, A-W, B, D.)

Religion: Hebrew (Memos A-S, A-T)

This is an odd mixture of positive and negative traits (with references to memos that they were obtained from), then immediately followed by the religion. It's hard, from the present vantage-point, to not see these things as indications of a latent anti-Semitism, or at least a tolerance of it.

That being said, other than in these kinds of files, the matter doesn't come up. And it was never invoked, on paper anyway, as a reason not to trust someone. To my knowledge no one was denied a clearance, or given unusual clearance trouble, merely because they were Jewish. However there were all sorts of clearance problems for people who had left-wing associations, and for a variety of reasons many people who were Jewish in the 1940s tended to have these sorts of associations (it is not a coincidence that many of the spies on the project were Jewish Communists from New York City — it was relatively easy to see Stalin and the Communists as being the main practical and ideological opposition to the Fascists for much of the 1930s). So it is not unreasonable to assume that for some security agents these were intertwined. Racism is less of a discrete variable than a sort of lens that is applied to everything, so I don't think it's beyond the pale to see it as a sort of "intensifier" for security clearance problems.

The immigrant question was in many ways more clearly defined, though. Immigrants from nations that the US was at war with — which after Pearl Harbor included Germany, Italy, Austria, Hungary, and many other places that had provided many of the refugee scientists — were judged very suspicious by the security forces. Enrico Fermi had huge problems getting full security access because of his Italian heritage (he was not allowed to travel freely for some time; he was confined to certain urban areas), and was accused of having fascist tendencies. Leo Szilard similarly.

In these cases, the security forces' excessively xenophobic judgments were not taken too seriously by the Manhattan Project leadership; they knew these people relatively well, knew they were refugees from Hitler, and knew they were working for the US effort. All of their complaints were eventually overruled and almost all of these scientists became naturalized US citizens during the war, which smoothed things over a bit.

Early on in the work, before it was formally the Manhattan Project, the Army and Navy handled all background clearance checks for the Office of Scientific Research and Development, and refused several clearances for important members on the basis of their alleged political backgrounds. Arthur Compton was one that the Navy wouldn't clear, because he had signed his name to one-too-many petitions over the years. The OSRD overruled this, and Vannevar Bush specifically noted that scientists were "queer types" who needed different standards than your regular candidates for military clearances, and also argued that for very important work, it was better to have them inside and monitored than outside and free.

General Groves similarly was willing to overrule his own (occasionally extremely over-zealous) security forces when it came to important people (notably Oppenheimer), with the argument being made that expedience took a priority over security (the only people he forcibly denied were ones directly suspected of espionage, notably a few students of Oppenheimer's from Berkeley). So in the end the Manhattan Project let a great many more "questionable" people work for it than the later Atomic Energy Commission would be allowed to, because of the time priority. It is an interesting statistic that percentage-wise, more women worked on the technical aspects of the Manhattan Project than later would work on the Apollo program — lending to the conclusion that it takes a certain amount of leisure to shoot yourself in the foot with prejudices...