Analysts at Alrajhi Capital said a merger might help both companies address their specific issues: declining operating profit in the past four quarters for Care and negative cash flow from operations in four of the five past quarters for Hammadi.
Alrjahi said if the combination goes through it would create the kindgom's largest listed healthcare provider.
Since the start of the year shares of both companies have under-perfomed their peers, mainly because of unsettled bills from the government.
Shares of airport ground handling service provider Saudi Ground Services fell 4.4 percent after it reported a 37.6 percent drop in second-quarter net income to 122.3 million riyals.
The index closed almost flat for the day and down 1.3 percent for the week.
In Abu Dhabi, shares of the largest listed developer, Aldar
Properties , climbed 1.3 percent to 2.37 dirhams after it reported a 5.6 percent drop in second-quarter profit to 620 million dirhams ($169 million), broadly in line with expectations.
Although revenue fell by a fifth from the year before, off-plan residential sales in the second quarter showed resilience in demand for mid-income housing, said a note by Arqaam
Government grants supported earnings growth in the second quarter, according to Arqaam, which has a positive outlook on project delivery.
"We view the momentum in real estate execution positively, in the run up to 2017 completions ... The handovers (2.4 billion dirhams in sales value), if realized this year, would trigger a discretionary dividend payment that is related to profit on residential handovers," said Arqaam Capital.
Arqaam rates the stock a buy with a target price of 3.4 dirhams.
The index fell 0.3 percent, as some shares that rose on the previous day fell on profit taking.
Dubai's index added 0.4 percent, its fifth straight session of gains as three of the five most valuable companies rose; Dubai Islamic Bank rose 1.5 percent.
Qatar's index rose 0.4 percent in very thin trade, half of the six listed banks rose, with Commercial Bank adding 1.5 percent.
In Egypt, shares of Global Telecom fell 0.5 percent after the company reported a second-quarter net profit of $35 million, growing by about a third from the year before.
"The reported numbers fall short from a profitability standpoint ... we infer positives from the management's tone in terms of outlook for Djezzy (the Alegerian unit) going forward; especially, taking positive note of the $150 million in dividends announced by Djezzy," said a note by Cairo's Naeem Brokerage.
The Cairo index closed near flat and down 1.4 percent of the week.
* The index added 0.07 percent to 7,086 points.
* The index added 0.4 percent to 3,675 points.
* The index edged down 0.3 percent to 4,596 points.
* The index rose 0.4 percent to 9,398 points.
* The index added 0.06 percent to 13,413 points.
* The index edged up 0.03 percent to 6,826 points.
* The index fell 0.3 percent to 1,323 points.
* The index lost 0.3 percent 5,066 points.
($1 = 3.7501 riyals)
(Editing by Larry King) ((email@example.com)(+971 5 6224 7653)(Reuters Messaging: firstname.lastname@example.org))