The pair had been on trial with Allan Johnston’s squad this week and have done enough to impress the Ayrshire club’s manager.
However, Kilmarnock now face a race against time to tie up the deals and get the duo registered in time to feature against Partick Thistle in Saturday’s Scottish Premiership clash.
Assistant manager Sandy Clark said: “We are talking to the players about getting them registered in time for Saturday but we will need to wait and see.
“We have made Ismael an offer and that has got to be negotiated. We are also checking the medical side of things to make sure we are okay from that aspect. Perhaps the biggest challenge is that his registration is still in Nigeria.
“If things go well, we might have him available for the weekend but there is a lot of work to be done yet.
“David Silva is another one we are talking to with the prospect of getting him on board. Whether that will happen before Saturday, I don’t know.”
Bouzid made 65 appearances for Hearts during two seasons at Tynecastle from 2009 to 2011, winning an international recall with Algeria.
The former Kaiserslautern and Galatasaray defender has since had spells in Greece, the United Arab Emirates and his homeland, while Silva left Kilmarnock last summer to play for Olhanense in his native Portugal.
However, ex-Rangers striker Nacho Novo and one-time Hull forward Tijani Belaid, who had also been hoping to win deals after agreeing to train at Rugby Park, will not be kept on.
“Nacho is not with us now,” added Clark. “We are looking at certain positions to fill but for Nacho’s position, we have got plenty of players who can fill that role.
“He came in and worked really hard, was a great lad and showed great attitude, but, unfortunately, we can’t take everybody.
“As for Tijani, he is a really good player but, again, it is an area where we are fairly well covered, although it’s one we may look at further down the road if he’s still available and we can move on a few.”