As stated, this question is not precise enough to have an answer, because it isn't posed in a way that it is accessible to more than vague intuition regarding what "omniscience" and "omnipotence" mean. A satisfactory definition must include a way to determine what can be expected of an omnipotent and omnipresent entity, and a procedure to find observable consequences of such an entity. If this cannot be done, the logical positist would say the question is meaningless in the sense of Carnap, an abuse of language.
Still, one can attempt to produce a more precise version of these things. When one does something like this, the result is often a logical contradiction.
Here is an example of a logical argument against omniscience:
"God doesn't know this statement is true." where by "this statement" I mean the very same statement in quotes.
If God knows this statement is true, then it is false, and God does not know it, in which case it is true.
Does God know that statement is true? Is this statement true or false?
(this is a variation on the well-known philosopher's sentiment "Searle cannot consistently believe this statement". Variations on this are found in Hofstadter's Metamagical Themas, "On Self Referential Sentences")
You can rephrase this as the religious doctrine of Maimonism. This is the main article of Maimonism:
"There is an omniscient God, who knows the truth or falsity of all religious doctrines, and, unfortunately for Maimonism, this God does not agree with the main article."
You can agree with Maimonism if you want, but God can't have a consistent opinion about it. In the preceding sentence, I assumed that the reader isn't God.
You might think that "this statement" is cheating. But it is easy to avoid using this construction, by a well known trick from computer science:
Consider the named strings A="consider the named strings",B="; then God doesn't know the fact asserted by the straightforward English meaning of the sentences formed by the concatenation of string A, an 'A', an equal sign, a quote, string A, a quote, a comma, a 'B', an equal sign, a quote, string B, a quote, and string B"; then God doesn't know the fact asserted by the straightforward English meaning of the sentences formed by the concatenation of string A, an 'A', an equal sign, a quote, string A, a quote, a comma, a 'B', an equal sign, a quote, string B, a quote, and string B."
This is a self-reproducing sentence. If you follow the instructions, you reproduce the very-same sentence that is giving you the instructions, and then the sentence asserts that the constructed sentence (itself) is not known by God, and the paradox is as before.
But this is not really a meaningful paradox the way I see it, in light of logical positivism. In order to make this fully meaningful, one needs to assume God's knowledge is queriable, so that the sentence has a meaningful procedure to determine truth or falsehood. If God is queriable regarding the truth of all sentences, then God becomes an oracle, and the halting problem with a queriable oracle is just as unsolvable as the halting problem without, so there are questions which the God oracle cannot answer using the procedure, at least not in finite time.
These paradoxes are due to Godel and Turing, and they are made precise using computer programs and purported oracles regarding the behavior of these computer programs. Such oracles cannot be computer programs, and if these oracles are realized in the physical world, they cannot be queried in finite time without leading to contradiction.
But the contradiction evaporates if you imagine that the oracle only knows at infinite time. So this is only saying that we cannot know God's opinion at finite time. This is analogous to the Catholic doctrine of "gradual revelation", that God's will is revealed more perfectly through time, and through the action of the holy spirit. So theologians do not have to worry about such contradictions, unless they wish to give meaning to "God does not know this sentence".
My opinion is that it is best not to worry about these paradoxes, and instead give a meaningful logical postivist definition for God that allows God to be subject to rational investigation. If this cannot be done, then God is not a meaningful concept. If it can be done, than the methods which are suggested by the positive definition for querying God can be used to determine the answer to all questions regarding God.