"It was possible for me to proceed to the US for my graduate studies because I was offered a scholarship". Is this sentence grammatically correct? Should it be "It became possible..... " Or are they both ccorrect?
"It became possible..." means that "it began to be possible" according to the definition of become from the OED. If the "scholarship" was what initiated the allowance of your "proceed[ing] to the US" then this would be a the more applicable wording; i.e. become would indicate a development of some kind and that the "scholarship" functioned as a contingency in determining the outcome of the situation while other factors are also present.
"It was possible..." suggest that the "scholarship" alone accounted for you ability to move "to the US." This also does not imply any change or development, whereas the word become does.
In this particular scenario, "became" would be the more suitable word; although both are grammatically correct. This is because at one point it was not possible for you to "proceed to the US" and through the development of the "scholarship" it "became" possible.