Praise be to Allaah.

Yes, it is permissible for him to join prayers, so he may join Zuhr and ‘Asr, and Maghrib and ‘Isha’, at the time of the earlier or later prayer, depending on what is easiest for him. Difficulty caused by sickness is one of the excuses which make it permissible to join prayers. The Prophet (peace and blessings of Allaah be upon him) granted a concession to the woman who suffered from istihaadah (non-menstrual bleeding), and he allowed her to join prayers. Narrated by Abu Dawood (287) and al-Tirmidhi (128);classed as hasan by al-Albaani in Saheeh al-Tirmidhi. 

Istihaadah is a kind of sickness, and Imam Ahmad quoted as evidence for it being permissible for a sick person to join prayers the fact that sickness is worse than travel. He was treated with cupping after sunset, then he ate supper then he joined Maghrib and ‘Isha’ prayers. End quote. Kashshaaf al-Qinaa’ (2/5). 

Note: 

It should be noted that the sick person for whom it is permissible to join prayers should offer each prayer in full without shortening it, because shortening the prayers is only permitted for the traveller. What some people think, that if a person joins the prayers at home because of sickness he is also allowed to shorten them, is not correct. 

Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said: The reason for shortening prayers is travel only, and it is not permitted when one is not travelling. As for joining prayers when one has an excuse or a need, if he needs to, he may join them when travelling short or long distances, and he may join them when it is raining and the like, or when he is sick and the like, or for other reasons. The aim behind this is to spare the ummah from hardship. End quote from Majmoo’ al-Fataawa (22/293).  

We ask Allaah to heal the sick Muslims and to bless them with patience and contentment, and compensate them with good. 

And Allaah knows best.