Both are correct. This is all about something called backshift where there is a shift from present tense to preterite in reported speech.
What he actually said was The project will be carried out, but I report this as He said the project would be carried out, using preterite would instead of present will.
In fact, even with preterite reporting verbs, backshift is often optional; you can keep the original present tense instead of backshifting it. Which is why will in your examples is perfectly acceptable.