John Lawler's comment above accurately describes the standard reach of project versus process, but writers often use project figuratively to describe things that aren't, in a literal sense, projects at all. Thus, from a 1941 translation of José Ortega y Gasset, Toward a Philosophy of History:
In spite of this, or rather because of it, technology is, strictly speaking, not the beginning of things. It will mobilize its ingenuity and perform the task life is; it will—within certain limits, of course—succeed in realizing the human project.
From "Spiritual Manifestations," in Southern Literary Messenger (July 1853):
There is one sufficient objection to the whole project of Spiritual Manifestations, as now obtruded upon us, though it is one apparently so simple that its real cogency will not be readily appreciated by all. That objection is that the theory is premature.
And from Juan Corradi, "The Avatars of Socio-Political Discourse in Latin America," in Social Science Information (1979) [combined snippets]:
Insofar as "catching up" with the achievements of research and development in advanced societies entails also a hierarchy of texts, a difference between primary and secondary sources, between production (that takes place primarily in central societies) and reception (the adaptation of scientific knowledge in Latin America), there was an apparent return to the principle of commentary and to the early mode of importation and diffusion, as if the oligarchic project of modernization, interrupted by populism, had to be restored. Yet this initial understanding of the process was deceptive. For oligarchic discourse was structured around a specific form of commentary: that of juridical exegesis.
So "project of modernization" is one turn of phrase available to you in English, though it uses "project" in a broad and somewhat fanciful way. On the other hand, as this Ngram chart for the years 1900 through 2008 indicates, "process of modernization" (the red line) is significantly more common in published English writing than "project of modernization" (the blue line):
To me, the two phrases have somewhat different implications. "The project of modernization" suggests a general and somewhat nebulous undertaking with no clear starting point and endpoint. "The process of modernization" suggests something that occurs in accordance with a preexisting analytical framework, with definite stages or steps. I don't know whether other native English speakers share my impressions of the two phrases, however.
In any event, with regard to your basic question "Which phrase shall I choose?" the answer is that both phrases make sense and may be used meaningfully, depending on what idea of movement toward modernization you're trying to express.