I have not found any mention of this being built in any Jewish or Israeli sources, but it would be between the Kotel and the Kotel HaKatan and 50-100 meters away from the Western Wall of the Temple Mount.
The Cotton Market was originally built in the 14th century and was for a time an important commercial center. But it fell into disuse and was abandoned for centuries, through Ottoman rule and through Jordanian rule.
Here's a picture of the market from National Geographic in 1917:
It was rebuilt only after Israel ruled the area, possibly as late as 1995, by the Waqf and paid for by the UN.
The Cotton Market mirrors the history of Jerusalem as a whole: when no Jews were there, no one cared about it or its historic value. Even when Jordan ruled Jerusalem nothing was done to rebuild this area. Only under Jewish rule was Arab interest rekindled in the Cotton Market and Jerusalem as a whole.
Remember this whenever you read an article about how sacred every stone of Jerusalem is to Muslims. The Cotton Market is undeniably a beautiful piece of architecture, ending in an impressive gate to the Temple Mount - but even so, it was abandoned and ignored by the very people who claim to love it the most today.
As with Palestine altogether, Arab and Muslim interest has traditionally been not so much in a homeland for Palestinian Arabs as in a place where infidels and dhimmis must not have any influence.
Here's a small experiment. Right now, the number of Palestinian websites that mention the Cotton Market (al-Qattanin) is relatively small. If in fact a synagogue will be built there, Google it in a few years and see how many websites will extol the supreme importance the Cotton Market has to Islamic history, all the while complaining about how the wicked "Zionists" have been working overtime to "Judaize" the city and threaten the mosques there.