Midfielder Clifford, 21, was released by Leicester at the end of last season, while 23-year-old striker Cowan has been without a club since leaving Dundee.
Bothmen have been on trial at Roots Hall will feature in the Shrimpers' friendly against Real Madrid C.
"Don has done well since he's been here and it will be good to have a further look at him in Madrid. He'll play some part in the game and we'll watch how he gets on in the training sessions," boss Phil Brown told the club's official website.
"Conor is a good footballer who for the last couple of years has been looking for a home. He's been with Chelsea and is the sort of midfielder I'm looking for at the moment - someone who will rough it up a bit when we need to."
However, fellow trialists Josh Rees, Mark Carrington, Marvin Wijks and Dan Fitchett will not make the trip to the Spanish capital.