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I have completed my Bachelor's degree (3 years) in Mathematics and pursuing MSc(2 years). Currently I am in final year and expect to obtain MSc degree in next May. I am willing to do my PhD (in Mathematics) in USA.
From the information obtained from my seniors and friends, I found that most of the USA institutions ask for GRE and TOEFL score. But are there universities of USA which do not need GRE and/or TOEFL score for admission in Phd in Mathematics?
The truth of the matter is that while you may find many schools that do not require the GRE, virtually every US school is going to require either the TOEFL or a similar test such as the IELTS.
The reason for this is that one of the important criteria for admission to most US graduate programs is the ability to do coursework and interact with fellow graduate students and faculty in their lingua franca, which in the US is English. Consequently, schools need to ensure that the students they accept have a certain level of proficiency in English before they enroll. (A number of grad school admissions officers have mentioned that they blacklisted candidates from certain countries because of problems related to fraud—candidates would hire other people to conduct telephone interviews with universities in the US, and the actual candidate would come with hardly any English whatsoever.)
So, for most schools, you will find it a requirement to be able to speak the dominant language of instruction—and not just in the United States. (For instance, all students at our university must show the ability to speak either German or English, depending on the program to which they apply.)
I think you should change your strategy for choosing a doctoral program. That is, determine which programs would provide you with the best training for the work you want to do. It might (probably will) involve these standardized test scores. You run the risk of narrowing your choices by starting with these types of requirements.
Also, you stated that you are "willing to do my PhD." Your decision for doing a PhD should be motivated by your ultimate career trajectory. I think you should clarify this career trajectory (i.e., Why do you want to do a PhD? Because it will prepare me for ...). Then, you should find a PhD program that would be best suited to help you establish this career trajectory. You don't want to get into a PhD program that is inappropriately matched with your trajectory, even though they don't require GRE and TOEFL scores. If you have scores that are not competitive, then it would be a good idea to focus on improving those scores.
Bottom line is you want good training.
To answer your question, no: all schools of reasonably well-known merit require a GRE. Shopping for schools that don't is only harmful. I am guessing your mindset is simply wondering if within, say, the top 100 programs there might be exceptions, perhaps because the test is a financial burden, but no, the GRE is the standard for admission. It's not the only important factor, but every school will require it. I've heard it described as a "sanity check" to make sure the student's abilities pass a check relative to what the recommendation letters and transcript advertise.
To flip the coin, imagine a school that doesn't require the GRE. Why not: what's wrong with it? Does it let students in who would have performed terribly, and risk a remedial first year? Does it let students in who are likely to drop out and simply don't fund them? Is it just really lacking merit? I would be extremely leery of such a school.
Regarding the TOEFL, the situation is simple. It would be foolish for a university to not make sure incoming students speak English. Even if you are sure of your ability and don't have the time to study or the money for the test, you should be leery of going to a school where there's a pretty good chance your classmates will include German, French, Japanese and Mexican students who don't actually speak English and went to the only school that admitted them after they failed the TOEFL.
On the other hand it is of course very important that you can pass. It would be miserable to marginally fail but not be able to teach a course effectively (which is how you pay for grad school in part) or collaborate with peers.
