The report by the Reuters news agency quoted an unnamed government official.
The figure is beyond the $2 billion (Sh176 billion) Kenya is seeking which will be used to finance infrastructure projects the current financial year.
The proceeds from sovereign bond will also be used to repay an existing syndicated Sh53 billion ($600 million) loan that matures this August.
A Eurobond is an international bond that is denominated in a currency that does not belong to the country where it is being issued.
According to analysts, the issue is attracting high demand because it comes in two tenors with a five-year tranche offering investors a minimum 6 percent return and a 10-year portion, guaranteeing a yield of 7 percent.
The Africa’s first sovereign dollar-debt was in Zambia which saw it raise $1 billion in April this year which was priced to yield 8.6 percent.
“Investors have a different appetite for debt depending on the duration and once the Eurobond market develops, you might find multi-tenor offers become more popular to cater for refinancing risk,” a fixed-income trader in Nairobi said.
Starting June 5 this year, Kenya kicked off an ambitious road-show campaign in the United States beginning San Francisco, Los Angeles then headed to Boston and New York.
The government had also planned to move to London and Germany in Europe starting June 11.
On Thursday last week Treasury Cabinet Secretary Henry Rotich presented the country’s budget for the 2014/2015 financial year which saw proceeds from the issue being included to finance the Sh342 billion deficit.
The first time Kenya had considered selling Eurobonds was in 1997.
The government forecasts economic growth will accelerate to 6.1 percent in the year through June 2015 from an estimated 5.3 percent in fiscal 2014, helped by exports of tea and flowers.