Moyes likes the 18-goal striker and considered an £8million offer in the January window.
He held off in favour of continuing his checks to see if Armstrong, 23, is worth the £10m-plus that Rovers want.
Other clubs are keen on Armstrong, including Fulham and Everton, but both see him as a squad man just now.
Rovers will sell Armstrong in the summer if they fail to go up.
Meanwhile, Andre Villas-Boas quit Marseille after they decided against signing West Ham midfielder Manuel Lanzini on loan.