They were among a total of 16 people who died in the region's prisons last year, according to data published by the Ministry of Justice.
Deaths by suicide were also recorded in Durham, Holme House and Low Newton prisons.
The Howard League for Penal Reform said across England and Wales, deaths by suicide have risen by 46 per cent in the last three years, at a time when prisons have struggled to cope with growing numbers of prisoners, overcrowding and staff cuts.
The charity, which works for less crime, safer communities and fewer people in prison, said a further eight prisoners were killed in apparent homicides during 2015 – the highest number in a single year since current recording practices began in 1978.
The Howard League and Centre for Mental Health, supported by The Monument Trust, are working together on a joint programme on preventing people from dying by suicide in prison.
Analysis of cases reported to the Howard League last year suggests that people who are held in prison on remand, or have been recalled to prison while on licence, are disproportionately likely to take their own lives.
It said about 40 per cent of prisoners who died by suicide in 2015 were on remand at the time of their deaths, even though people on remand make up only 15 per cent of the prison population at any one time.
About 14 per cent of prisoners who died by suicide in 2015 were in prison due to a licence recall. Recalled prisoners account for only seven per cent of the prison population.
Frances Crook, chief executive of the Howard League for Penal Reform, said: "No one should be so desperate whilst they are in the care of the state that they take their own life.
"The numbers hide the true extent of misery for prisoners and families – and for staff, who have been given the impossible task of keeping people safe in overcrowded prisons starved of resources.
"The question now for the Ministry of Justice is: what to do? This level of deaths, violence and anguish in prisons cannot continue to rise in a civilised society. We cannot go on cramming more people into jails without any thought for the consequences."