"Ministers typically send letters of expectation to the boards of entities in their portfolio. This letter was prepared after the Treasury identified an opportunity to bring the accountability framework into line with other Crown agencies," English said in a statement.
"This change reflects the government's focus on good public sector practice and has not been driven by any concerns with the Reserve Bank's performance."
Inflation accelerated at a lacklustre 0.4 per cent annual pace in the first three months of this year, and has now been below the target band for each of the past six quarters.
The Reserve Bank has said it can tolerate an extended period of weak inflation because it views the phenomenon as temporary, driven by a slump in oil prices. It's also wary that cutting interest rates further to bolster inflation could inflame the housing market where it has concerns about inflated prices causing financial instability.
English told BusinessDesk last month that he was happy the Reserve Bank was acting consistent with the agreement, and acknowledged they were in "some challenging territory".
The Reserve Bank's move to introduce lending limits to curb price gains in Auckland's housing market under its financial stability mandate has also sparked speculation of tensions between it, Treasury and the government.
In his November letter, English noted he wanted the board to monitor the bank's important stakeholder relationships, with himself, Treasury, regulated entities and with other agencies, and keep under review how the relationships are operating in practice.
Governor Wheeler's five-year term ends September 2017 and he hasn't yet indicated if he will seek a second term. Governors are appointed by the Minister of Finance on the recommendation of the Reserve Bank board.
Asked by BusinessDesk in an interview last month how likely it was that Wheeler might stay for another term, English wouldn't comment beyond saying that would "unfold". Questioned on tensions over low inflation and housing, English said: "we are not at all concerned about those things".