That's despite many large mutual and pension funds remaining on the sidelines of the scheme due to regulatory and technical problems.
At the current pace of usage, and as more firms come on board, this quota will be filled by around the end of the first quarter, Hong Kong exchange executive Tae Yoo told an industry seminar hosted by the Irish Funds Industry Association (IFIA) in Hong Kong last month.
A trading uptick in late December and early January means quota take-up has grown increasingly active, said Yoo, who oversees fixed income currency and client business development.
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"At this pace, if it continues, by end of first quarter, in two or two-and-a-half months, you're looking at a full quota," he said.
Yoo added that the exchange was in a dialogue with regulators and the Shanghai bourse over the quota and that he anticipated discussing whether to raise or even remove it altogether when the quota was close to being reached.
If the limit is not raised, investors will only be able to buy more Shanghai shares through Stock Connect as and when other firms free-up the quota by selling stocks.
Given the hesitant start of the trading scheme, few investors expected that limit would be reached so soon.
Responding to a request for comment by Reuters, a spokesman for the Hong Kong exchange said it "monitors quota use and will discuss the issue with the relevant authorities at the appropriate time."
Hitting the limit for so-called Northbound purchases could be problematic for Hong Kong-based foreign investors, as they have also nearly exhausted a 270 billion yuan quota granted to them under a parallel cross-border investment scheme known as RQFII.
The Hong Kong exchange has said it regards the Stock Connect quota as a speed bump that can be removed over time, but the final decision will likely rest with China's State Administration of Foreign Exchange (SAFE), which has been slow to raise quotas in the past.
SAFE could not be reached for comment.
Regulatory hurdles have so far kept the largest global investors out of the Stock Connect scheme, suggesting demand for the Northbound quota, which is limited to 13 billion yuanper day, could explode once these players come on board.
Dean Chisholm, head of operations for Asia Pacific at asset manager Invesco, said a daily quota squeeze would make it hard to launch investment products targeting Chinese equities.
"People who need access to liquidity on a daily basis, for example exchange-traded-funds, will be very hesitant about using Stock Connect," he said.
Some foreign institutional investors have already been pushing for the Stock Connect limit to be raised, according to industry insiders. They fear the quota will be almost used up by the time they have won the green light from their home regulators to use the scheme.
"Demand from managers is clear and the IFIA is addressing the remaining issues with the relevant authorities," said Pat Lardner, CEO of the IFIA.
"Ensuring sufficient quota to meet the demand we expect is therefore also essential."