India has provided billions of dollars of aid to Afghanistan since the fall of the Taliban in 2001, and is keen to protect against the return of a radical Islamist regime in Kabul after international troops pull out in 2014.
But any Indian activity in Afghanistan triggers sensitivities in neighbouring arch-rival Pakistan, which fears losing influence in Afghanistan and being encircled.
In October last year, India and Afghanistan signed a "strategic partnership" aimed at deepening security and economic links, with Karzai keen to elevate India's involvement further.
At a meeting on Saturday with business leaders in Mumbai, Karzai promised a "red carpet" welcome for Indian investment as US-led NATO forces prepare to withdraw.
"Indian businesses need not be shy while thinking about Afghanistan," he said. "Chinese businesses were there long before you came, five or six years ago.
"If you don't arrive on the red carpet, it will get dusty. Therefore, do hurry up in coming and take advantage of the opportunities in Afghanistan."
India has not contributed to the international coalition of forces deployed in Afghanistan, but has worked on low-key troop and police training as well as infrastructure development.
The US was previously wary of India's presence in case it led to a "proxy war" with Pakistan on Afghan territory, but US Defense Secretary Leon Panetta visiting New Delhi in June encouraged India to play a more active role.
At a meeting soon afterwards to promote trade with Afghanistan, then Indian foreign minister S.M. Krishna said potential business interests must overcome "the anxiety of withdrawal, uncertainty, instability and foreign interference".
Last year a consortium of Indian companies won the rights to develop Afghanistan's largest iron ore deposits, underscoring growing ties between the two countries.
Afghanistan is believed to have mineral reserves worth at least $3 trillion which could generate billions of dollars in tax revenue.
Karzai, who has been in power since 2001, is expected to sign fresh mining deals in New Delhi, as well as pacts on youth affairs, development projects and fertiliser trade.
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So this is what india always does... Without killing a single person, it wins people's hearts and trust, and then ultimately their resources and then later, their man power... God bless my country and its people, and so does it blesses afghans for letting us use their minerals...
Ohhh wow! So finally India's soft power is emerging in a extremist nation as afghanistan, that too without sending a single soldier there... Proud of my nation's foriegn policy!!! I hope india and afghanistan together arrive as a friendly nation so that we indians can extract afghani minerals in a such a way that there is no tomorow:p without even hurting a single afghani...