The former Gunners captain, 64, and assistant boss to Wenger since 1996, is undergoing treatment in hospital.
Belfast-born Rice started his career in the Arsenal academy and went on to lead the side in the 1979 FA Cup final win over Manchester United before after making some 528 appearances for the Gunners, the Northern Ireland defender finished his playing career at Watford.
He received an MBE from the Queen in 2013 ahead of his retirement as Arsenal assistant manager at the end of last season, having been persuaded to stay on in the dugout by Wenger.
Thousands of Arsenal fans took to social media sites to offer their support for the club legend following confirmation of his illness, and Wenger wishes his good friend a speedy recovery.
Speaking to the club's official website, www.arsenal.com, Wenger said: "It is very sad news. Not only do we love him, but of course we are very grateful for what he has done here.
"We are in touch with him, but I trust his legendary strength to get over it.
"I am confident that he still has the fighting spirit he always had to get over it and that he will come out in a successful way."
Wenger continued: "That is what he needs now (strength of character) - he has that. In such a difficult period, it is a huge resource.
"His mental state is very determined, very strong as he always has been.
"He will fight for that, I have complete trust that he will come out in a very strong way and of course let us hope for the best."
On the pitch, Arsenal will be out to consolidate their place at the top of the Barclays Premier League when they travel to Cardiff on Saturday.
The Gunners beat Marseille 2-0 on Tuesday night to all but secure their place in the last 16 of the Champions League.
Left-back Kieran Gibbs missed the European tie through illness, but Wenger revealed he would be back in contention this weekend.
The 24-year-old returned to the England senior squad over the international break and has been enjoying a good run of form.
Gibbs has been through his share of fitness problems, following a broken metatarsal in November 2009, not to mention a double-hernia operation and in January this year a torn thigh muscle sidelined him for some two months.
The defender's consistency, having started all 12 Premier League matches so far, is keeping Spain international Nacho Monreal in reserve.
With Everton's Leighton Baines facing an injury lay-off himself, now could be the perfect time for Gibbs to set down a marker for the 2014 World Cup.
"I think this year I have moved up a level. I have stayed fit and finally shown what I can do," said Gibbs.
"Hopefully I can stay fit as possible and play as much as I can.
"I feel strong and I have taken a good look at myself physically.
"I have done a lot of gym work and preparation and even after games I have worked hard. I have to stay ahead of everything and make sure I don't fall behind or pick up any silly injuries.
"I am more relaxed knowing I can trust my body. You can get on with the game and that is all I wanted to do.
"The games will come thick and fast now all the way through the winter, that is going to be an important test for me."
To come through such a hectic schedule unscathed and with continued form on into the spring, Gibbs could well move a step closer to booking a summer ticket to Brazil.
"England was a big positive for me, especially as Roy Hodgson had three left-backs in the squad, but he told me that he felt he had to bring me in because of the way I have been performing," Gibbs added in the official Arsenal matchday programme.
"That gave me a lot of confidence. I managed to sneak on against Germany as well and get another cap.
"I am really happy to be back involved in the set-up and it is obviously a good time to be back in the squad in World Cup year."