The West Indies were 4-318 at stumps on Sunday (Monday AEST) on the third day of the Second Test, in reply to Sri Lanka's 9-549 declared.
On a day of measured progress, only three wickets fell and 260 runs were added. Only 13 wickets have fallen in the first three days.
Hope was 86 not out and Greaves 85 not out at the close of play, approaching their fifth and third Test centuries respectively.
The pair also were closing in on their highest partnership of 196, achieved against New Zealand in Christchurch in December.
The West Indies lead the two-Test series after winning the first Test by an innings and 217 runs.
With that in mind, the onus was on Sri Lanka to make the play on the third day but wickets were hard to come by on a placid pitch at the Sir Vivian Richards Stadium.
Only one wicket fell in the first session on Sunday as John Campbell (72) and Kavem Hodge (31) put on 89 for the second wicket after West Indies resumed at 1-58.
Prabath Jayasuriya then removed Campbell and Amir Jangoo (9) in the space of around four overs to bring Hope and Greaves to the crease.
West Indies' innings was in the balance at 4-144 but Hope and Greaves defied the Sri Lanka bowlers for the rest of the day and by stumps West Indies trailed by 231 runs.
Hope reached his half century from 90 balls with five boundaries and Greaves followed him to that milestone from 85 balls with six fours. Jayasuriya had 2-77 from 36 overs after a day of hard work.
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