Kenya's banking industry is undergoing consolidation after the closure of three lenders last year exposed weaknesses caused by lapses in corporate governance.
The government also capped commercial lending rates and set a minimum interest rate last September, further squeezing banks' margins and putting pressure on lenders to consolidate.
NBK, in which the government and the state pension fund NSSF have a combined 70 percent stake, has been particularly hard hit, sliding into the red and falling below statutory minimum capital requirements.
Both banks have the government and the NSSF as their largest shareholders and they have traditionally competed for public sector banking business, including deposits.
KCB is casting the takeover offer as a solution to the crisis at NBK, which has eroded shareholder value of what was previously one of Kenya's biggest banks, the documents showed.
If the deal goes ahead, the combined bank will have assets of 750 billion shillings ($7.3 billion), with the capacity to grow to 1 trillion shillings within three years, the documents seen by Reuters on the proposal showed.
The value of the deal will be based on the market valuation of NBK. Analysts said it was hard to gauge the eventual price since NBK has some preference shares, which would have to be converted into ordinary equity at an agreed rate.
Reuters on Monday that the bank had sent an expression of interest in acquiring a controlling state in NBK to the finance ministry. He did not offer more details.
NBK's shares, which hit a 14-year low of 5 shillings ($0.0484) in March, jumped as much as 10 percent on Monday to 7.7 shillings on news of the takeover bid.
Francis Mwangi, a banking analyst at Standard Investment Bank, said the deterioration of NBK's financial performance had made it an attractive acquisition target.
"It always makes sense to acquire a company that is struggling because of assets that can be turned around. That's how you unlock value in a short period of time," Mwangi said.
KCB is open to other takeover scenarios than its proposed share swap, the documents showed.
In its proposal, KCB, which also operates in neighbouring countries, said it would initially take over 70 percent of NBK shares, before announcing its offer for the remaining stake.
Other deals announced in Kenya's banking sector in recent months include the purchase of tiny lender Fidelity by the State Bank of Mauritius, and the acquisition of Habib Bank Kenya by Diamond Trust Bank in a share swap.