This manager's behavior has insecurity stamped all over it. Unfortunately, you cannot change the behavior of an insecure manager by issuing an order. In fact, by telling him to stop asking if he should resign, you may have actually fueled his insecurity because he is unable to follow your directive.
The first action to take is to try to determine if there is anything about your leadership style or developments in the company itself that may be exacerbating his feelings of insecurity. If there are threats of layoffs or impending downsizing, or if your style is one of those "my-way-or-the-highway" approaches, you can expect him to continue to quiver.
Although the era of making grandiose promises to employees about their future with a company has passed, there are some steps that you can take to help this manager feel more secure. For example, since insecurity can be heightened because of a lack of information about what is going on in the company, you should consider having increased communication with him about plans, changes, and developments. Let him know more about the near-term and longer-term goals and the strategies to meet them, and try to give him more of an opportunity to express his ideas and suggestions. The more he knows about what is going on, the more secure he is likely to feel.
In addition, increased communication with this manager will also provide him with more of an opportunity to discuss his work with you, and this should lead to a decrease in mistakes and a further increase in his feelings of security.
At the same time, it will be important for you to continue to monitor his mistakes, particularly in terms of their magnitude, frequency, and the extent to which he is learning from them. Unless he starts to show some improvement, you may want to consider his resignation offer more carefully.
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