Until recently, Kenya was not in a position to go after the business. Its digital infrastructure - and particularly its satellite communications links with other countries - was prohibitively expensive for firms that rely on instant communication.
That changed in 2009, when the first of three undersea internet cables arrived in the country.
"I travel around the world trying to make the case for international investment in outsourcing," says Mr Kukubo.
"Before the cables came it wasn't even a discussion. Now we have seen a huge interest with companies saying 'you have the basic ingredients'."
Mr Sikka, whose firm initially used a grant from the World Bank to help pay for the expensive satellite links, agrees.
"It's had a huge impact on cost," he says. "It's made Kenya competitive with the rest of the world."
"Our bandwidth costs have dropped about 90%. However, the part that is most exciting to us is that we are able to go out and get higher value more attractive work because we have the fibre optic capability, which pays us more."
He says the cables "offer a perception of reliability" to potential customers.
"Previously, the satellites offered that reliability, but not the perception," he says.
But reliable cheap bandwidth is not the only selling points for Kenya, says Mr Sikka.
He highlights the country's "abundant talent pool" and its "neutral accent".
"India invests a lot in so-called accent neutralisation, where the agents are taken through a course," says Mr Sikka, himself a veteran of the Indian outsourcing scene.
Kenya, he says, does not have that problem. It's a mantra repeated by everyone pushing Kenya as an outsourcing destination.
Mr Sikka also emphasises the population's "cultural affinity" with countries such as the US and the UK.
"Cultural affinity to the UK is very important," he says. "It is very important to show an empathy and understanding of the culture. It can't be someone calling up from an alien world with no understanding about what it means to live there."
He says young Kenyans consume the same TV shows and music as their counterparts in the US and the UK, and they follow the same sports teams.
The legacy of the British schooling system in Kenya is also an advantage, he says.
"The main thing when you are talking on the phone is getting into the customer's shoes."
For example, the firm does processing work for an unnamed smartphone application that allows users to take a picture of a product to find the best prices online.
The pictures are sent to a team of people in their offices, who analyse the images and tag them with descriptive words that can be used to find the best prices on the web.
"The results are sent back to the phone within 30 seconds," says Lakshmanan Manickam, general manager of the firm, and another person who has moved from India to kick-start the industry.
This work requires a "highly-skilled workforce", he says, who are able to describe objects accurately.
The firm also creates content for websites and audits transcriptions and translations for US companies.
He says it is much easier - and cheaper - to set up these projects in Kenya; factors that will make it "the next outsourcing destination".
But not everyone is so optimistic about Kenya's chances.
"I'm more conservative about what it can do," says Erik Hersman, an entrepreneur and a major figure in the local technology start-up community.
"If you are just trying to bid against your Indian counterparts, I think it is a hard thing to do, particularly when they have so much experience."
He believes that rather than offering similar services to other countries, Kenya needs to find its own niche if it is to be successful.
"I think it is a little bit more of a difficult space than people make it out to be."
In addition, he says, the country should not try to compete with other countries on price.
Instead, he says, firms and government should look at the kinds of graduates that are coming out of the country's universities and offer services based on their skills.
This has worked for countries such as Bangladesh, he says, which has developed a concentration of animation firms.
He says Kenya's historical ties with the UK, which means it has similar accountancy and legal systems, offers opportunities for outsourcing firms.
However, Mr Hillary says his studies conducted with the United Nations suggest the country may face other challenges.
"We found the biggest inhibitor was people's sense of corruption," he says. "But, it's more about the perception of doing business than the reality."
A spate of violence following the disputed 2007 presidential elections also damaged the country's reputation.
But the government's Mr Kukubo remains optimistic.
"One has to be candid about these things," he says.
"[The 2007 violence] was one of those things that was incidental and rare in Kenya's history.
"Out of that crisis was born a new Kenya, really. Bad as it may have been, I think we learnt out lessons."
In the next year, he says, there will be an "explosion of interest" from firms wanting to set up in Kenya.
It has already invested in a new business park on the Mombasa Road, a stone's throw from Kencall, Horizon and Ken-Tech data.
The 500,000sq ft (46,450sq m) of empty office space, primed for an influx of local and India firms, is an indication of its conviction.
And it is a validation for Kencall's Mr Nesbitt.
"When we started Kencall, we were really a voice in the wilderness talking about outsourcing. I think everyone sat there with one raised eyebrow waiting to see what could happen.
"But people have now begun to see that this can actually work in Kenya."