The women were receiving a fertility treatment called intra-cytoplasmic sperm injection, or ICSI, in which a single healthy sperm is injected into a woman's egg with a pipette.
The technician performing the injections is believed to have changed the pipette after each procedure, but not the rubber top - which should have been protected by a filter.
The technician is believed to have raised the alarm after discovering traces of sperm in the rubber top.
About half of the 26 have already become pregnant or given birth, the clinic said.
"For some of the 26 couples, frozen embryos are still available but the chance remains that they [too] have been fertilised by the sperm from a man other than the intended father," the UMC said.
The couples have been informed.
"The UMC's board regrets that the couples involved had to receive this news and will do everything within its powers to give clarity on the issue as soon as possible."
ICSI is one form of in vitro fertilisation (IVF), which involves a series of procedures used to treat fertility or genetic problems and assist with conception.
During IVF, mature eggs are collected from a woman's ovaries and fertilised by sperm in a laboratory.
The fertilised egg (embryo) or eggs are then implanted in the uterus. One cycle of IVF takes about two weeks.
The technique has allowed older women and couples with fertility problems that would normally prevent them having children to become parents.
The technology has seen significant advances in recent years, and is now considered reliable enough that one British firm is offering a "baby or money back" scheme to potential clients.
But mix-ups do occur.
In 2012 a Singapore mother sued a clinic for alleged negligence after it mixed up her husband's sperm with that of a stranger. In that case, the ethnic Chinese woman first suspected that something was amiss when her baby, who was born in 2010, had markedly different skin tone and hair colour from her Caucasian husband, news reports at the time said.
In 2014, an Italian woman became pregnant with another couple's twins after a mix up at a clinic in Rome. She and her partner decided to continue with the pregnancy.
An investigation into British IVF clinics in the same year revealed three similar "grade A errors" between 2009 and 2012. Incidents included a couple being given the wrong sperm, resulting in a child being born with a different genetic father to its sibling.
Another incident listed in the 2014 report involved dishes with the embryos of 11 patients becoming contaminated with "cellular debris that may have contained sperm". Most of the incidents the report recorded at British clinics were classed as less serious B and C-class blunders, including breaches of confidentiality and eggs becoming unusable after being extracted.
In October, a study showed male children from one form of IVF are more likely to have low sperm counts, possibly reflecting inherited fertility problems.
